New Trends in Integrability and Partial Solvability

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Series II: Mathematics, Physics and Chemistry – Vol. 132

New Trends in Integrability andPartial Solvability

edited by

A.B. ShabatLandau Institute for Theoretical Physics,Russian Academy of Science, Moscow, Russia

A. González-LópezDepartamento de Física Teórica II,Universidad Complutense de Madrid, Spain

M. MañasDepartamento de Física Teórica II,Universidad Complutense de Madrid, Spain

L. Martínez AlonsoDepartamento de Física Teórica II,Universidad Complutense de Madrid, Spain

and

M.A. RodríguezDepartamento de Física Teórica II,Universidad Complutense de Madrid, Spain

Springer-Science+Business Media, B.V.

Proceedings of the NATO Advanced Research Workshop onNew Trends in Integrability and Partial SolvabilityCadiz, Spain2–16 June 2002

A C.I.P. Catalogue record for this book is available from the Library of Congress.

Printed on acid-free paper

All Rights Reserved© 2004

No part of this work may be reproduced, stored in a retrieval system, or transmittedin any form or by any means, electronic, mechanical, photocopying, microfilming,recording or otherwise, without written permission from the Publisher, with the exceptionof any material supplied specifically for the purpose of being enteredand executed on a computer system, for exclusive use by the purchaser of the work.

ISBN 978-1-4020-1836-7 ISBN 978-94-007-1023-8 (eBook)

DOI 10.1007/978-94-007-1023-8

Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2004 Softcover reprint of the hardcover 1st edition 2004

Contents

Preface

List of Contributors

M.J. Ablowitz and J. Villarroel/ Initial value problems andsolutions of the Kadomtsev–Petviashvili equation 1

F. Calogero/ Partially superintegrable (indeed isochronous)systems are not rare 49

A. Degasperis, S.V. Manakov, P.M. Santini/ Initial-boundaryvalue problems for linear PDEs: the analyticity approach 79

S.N. Dolya and O.B. Zaslavskii/ Quasi-exactly solvable Bosesystems 105

V. Dryuma/ The Riemann and Einstein–Weyl geometries in thetheory of ordinary differential equations, their applicationsand all that 115

F. Finkel et al./ Dunkl operators and Calogero–Sutherland models 157

V.M. Goncharenko and A.P. Veselov/ Yang–Baxter maps andmatrix solitons 191

P.J. Olver/ Nonlocal Symmetries and Ghosts 199

S.N.M. Ruijsenaars/ Integrable BCN analytic difference operators:hidden parameter symmetries and eigenfunctions 217

A.B. Shabat and L. Martınez Alonso/ On the prolongation of ahierarchy of hydrodynamic chains 263

P. Winternitz/ Superintegrable systems in classical and quantummechanics 281

vii

xi

Preface

The Advanced Research Workshop “New Trends in Integrability and PartialSolvability” (ARW.978791) took place in the beautiful setting of the Facultyof Medicine of Cadiz University’s main room on June 13–15, 2002. Althoughthe number of participants was 30, the lectures were attended by more thanone hundred researchers from around the world who were also attendingthe NEEDS 2002 meeting. The aim of the organizers was to take advantageof these events to bring together researchers from the field of integrablesystems and/or from the particular subject of partial integrability, in viewof the current interest in combining methods and ideas arising from bothareas. A wide variety of topics were covered in the talks and the subsequentdiscussions, including the analysis of reductions and solutions of integrablenonlinear partial differential equations and dynamical systems, new meth-ods for the analysis of initial-boundary value problems for linear partialdifferential equations, quasi-exactly solvable Bose systems, the geometrictheory of ordinary differential equations, exactly and partially solvable spinmodels, the theory of nonlocal symmetries of differential equations, andsuperintegrable systems. The workshop revealed the growing importance ofthe theory of integrable system as well as the emerging theory of partiallysolvable systems.

The present volume contains a series of invited contributions describingthe background and recent developments of the main subjects discussed inthe workshop. Special emphasis has been laid on providing self-containedand detailed presentations of the theory.

M.J. Ablowitz and J. Villarroel give a detailed description of the inversescattering for the KP equation, a keystone in the theory of integrable sys-tems. The authors investigate how the inverse scattering method should beapplied to obtain solutions decaying off a background line, multi-pole lumpsoliton solutions and slowly decaying solutions.

F. Calogero presents some recent work on dynamical systems such thatall solutions with initial conditions in a certain region of the phase space areperiodic and have the same period. Many examples of this kind of systems,which the author calls “partially superintegrable”, are discussed in detail.

A new method for dealing with initial-boundary value problems for lin-ear PDEs, the analyticity approach, is systematically presented in the con-tribution by A. Degasperis, S.V. Manakov and P.M. Santini. The methodis applied to several classical problems (Dirichlet, Neumann, mixed, peri-odic) for second- and third-order evolutionary PDEs in 1 + 1 and in n+ 1dimensions.

S.N. Dolya and O.B. Zaslavskii consider the extension of the conceptof quasi-exact solvability to Bose systems. They find conditions for an

vii

PREFACE

algebraization of part of the spectrum and, in some cases, explicit formulaefor several energy levels of an anharmonic Bose oscillator. Two importantaspects of this contribution are, on the one hand, that the results do notdepend on perturbation techniques and, hence, are valid in the strongcoupling regime and, on the other hand, that they can be extended tomany-particle Bose systems with interaction.

V. Dryuma focuses on the study of the geometric notions underlyingcertain types of second-order scalar ordinary differential equations. A familyof second-order differential equations polynomial in the first derivative isstudied by relating it to the equations of the geodesics of a four-dimensionalRiemannian metric of a certain kind. The so-called dual equation of anarbitrary second-order differential is also investigated, and its relation tothree-dimensional Einstein–Weyl spaces is elucidated.

A general method for constructing and classifying exactly or quasi-exactly solvable Calogero-Sutherland models is discussed in the article byF. Finkel, D. Gomez-Ullate, A. Gonzalez-Lopez, M.A. Rodrıguez and R.Zhdanov, whose main ingredient is the use of new types of Dunkl operators.The authors obtain, in particular, several families of quantum spin modelswith elliptic interaction.

The contribution by V.M. Goncharenko and A.P. Veselov describes someof the authors’ latest results on the quantum Yang–Baxter equation. Theroad for these new findings departs from the inverse scattering theory of thematrix Korteweg–de Vries equation. In particular, the authors show howprevious formulae for the two soliton solutions of the matrix Korteweg–de Vries equation can be extended to give new Yang–Baxter maps on theGrasmannian G(k, n) and on the Cartesian product of two GrasmanniansG(k, n)×G(n− k, n), and find the corresponding Lax pairs for such maps.

The theory of nonlocal symmetry algebras is the subject of the contribu-tion presented by P.J. Olver. A new and consistent framework for the lattertheory is developed including, in particular, a description of the calculus of“ghost vector fields”. In this way the apparent outstanding paradox of theviolation of the Jacobi identity by certain nonlocal vector fields is resolved.

S.N.M. Ruijsenaars’ contribution uncovers the deep structure of theBCN relativistic Calogero–Moser model, whose basic operators are analyticdifference operators of D4 or D8 type, depending on the hyperbolic orelliptic character of the model. The author develops a “relativistic” versionof the hypergeometric functions possessing most of the usual properties oftheir classical counterparts.

A.B. Shabat and L. Martınez Alonso present a prolongation of a hier-archy of hydrodynamic chains which exhibits a universal character, as itreduces to many of the standard integrable hierarchies. Several new inte-grable nonlinear models are derived and the properties of the differential

viii

PREFACE

reductions of this enlarged hierarchy are studied.P. Winternitz presents some new insights into the theory of superinte-

grable systems, specially in relation to the existence and practical compu-tation of third-order integrals of motion.

The workshop was mainly financed by a grant of the Cooperative Scienceand Technology Sub-Programme of the NATO SCIENCE PROGRAMME.We also received financial support from the Complutense University ofMadrid and the University of Cadiz. We would like to thank the authoritiesof the University of Cadiz, where the workshop took place, for generouslymaking several of the university’s facilities available to the participants.

Finally we would like to thank Ms. Asuncion Iglesias for her excellentwork with the administrative arrangements during the workshop.

Madrid, June 2003

The Editors

ix

List of Contributors

M.J. Ablowitz

Department of Applied Mathematics University of Colorado, Boulder 80309-0526, USA

F. Calogero

Dipartimento di Fisica, Universita di Roma “La Sapienza”, 00185 Roma,Italy, and Istituto Nazionale di Fisica Nucleare, Sezione di [email protected]

A. Degasperis

Dipartimento di Fisica, Universita di Roma “La Sapienza”, Roma, ItalyIstituto Nazionale di Fisica Nucleare, Sezione di [email protected]

S.N. Dolya

B. Verkin Institute for Low Temperature Physics andEngineering, 47 Lenin Prospekt, Kharkov 61164, [email protected]

Valerii Dryuma

Institute of Mathematics and Informatics, AS RM, 5 Academiei Street,2028 Kishinev, [email protected]

F. Finkel

Dpto. de Fısica Teorica II, Universidad Complutense, 28040 Madrid, [email protected]

D. Gomez-Ullate

Centre de recherches mathematiquesUniversite de Montreal, C.P. 6128, succ. Centre-Ville,Montreal, QC, H3C 3J7, [email protected]

V.M. Goncharenko

Chair of Mathematics and Financial Applications, Financial Academy, Leningrad-sky prospect, 49, Moscow, [email protected]

A. Gonzalez-Lopez


xi

LIST OF CONTRIBUTORS

L. Martınez Alonso

Departamento de Fısica Teorica II, Universidad ComplutenseE28040 Madrid, [email protected]

Peter J.Olver

Department of Mathematics, University of Minnesota,Minneapolis, MN, USA [email protected]

M.A. Rodrıguez


S.N.M. Ruijsenaars

Centre for Mathematics and Computer Science,P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

P. M. Santini

Dipartimento di Fisica, Universita di Roma “La Sapienza”, Roma, ItalyIstituto Nazionale di Fisica Nucleare, Sezione di [email protected]

A.B. Shabat

Landau Institute for Theoretical Physics, RAS, Moscow 117 334, Russia

A.P. Veselov

Department of Mathematical Sciences, LoughboroughUniversity, Loughborough, Leicestershire, LE 11 3TU, UKLandau Institute for Theoretical Physics, Kosygina 2, Moscow, 117940,[email protected]

J.Villarroel

Universidad de Salamanca, Facultad de Ciencias, 37008 Plaza de la Merced,Salamanca, Spain

P. Winternitz

Centre de recherches mathematiques etDepartement de mathematiques et statistique,Universite de Montreal, C.P. 6128, succ. Centre-Ville,Montreal, QC, H3C 3J7, [email protected]

xii

S. V. [email protected]

Landau Institute for Theoretical Physics, Moscow, Russia

LIST OF CONTRIBUTORS

R. Zhdanov

Dpto. de Fısica Teorica II, Universidad Complutense, 28040 Madrid, Spain

xiii

O.B. Zaslavskii

Department of Mechanics and Mathematics, KharkovV.N. Karazin’s National University, Svoboda Sq. 4,Kharkov 61077, [email protected]

INITIAL VALUE PROBLEMS AND SOLUTIONS OF THE

KADOMTSEV–PETVIASHVILI EQUATION

M.J. ABLOWITZDepartment of Applied Mathematics University of Colorado,Boulder 80309-0526, USA

J. VILLARROELUniversidad de Salamanca, Facultad de Ciencias, 37008 Plazade la Merced, Salamanca, Spain

Abstract. Initial value problems and solutions associated with the Kadomtsev–Petviashviliequation are analyzed. The discussion includes the inverse scattering transform for suit-ably decaying data, solutions decaying off a background line, multi-pole lump solitonsolutions and solutions which are slowly decaying. Existence and uniqueness of theassociated eigenfunctions are discussed in terms of natural functional norms.

1. Introduction

The Kadomtsev–Petviashili (KP) equation

(ut + uxxx + 6uux)x + 3ε2uyy = 0 , (1.1)

with both ε2 = 1 or −1, is a physically significant equation which arises inthe study of small amplitude, long, two dimensional surface water waveswith surface tension, which vary slowly in the transverse direction to thewave propagation [1]. It is also an important equation in plasma physics[2]. Corresponding to ε2 = −1 we speak of KPI, while KPII corresponds totaking ε2 = 1. The KP equation is an extension to 2 + 1 dimensions of theprototype equation integrable by the inverse scattering transform (IST) in1+1 dimensions: the Korteweg–deVries (KdV) equation, which arises undersimilar assumptions. Its importance is highlighted by the fact that it is thebest known prototype of a 2+1 dimensional integrable equation. This meansthat the solution of the corresponding Cauchy problem with decaying datain the plane can be obtained by means of the inverse scattering transform.In [3–5] the relevant extension of IST to the multidimensional case was de-veloped and used to find the solution of the corresponding Cauchy problem

1

A.B. Shabat et al. (eds.), New Trends in Integrability and Partial Solvability, 1–47.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

2 M.J. ABLOWITZ AND J. VILLARROEL

with decaying data for eqs. (1); we note that IST has been henceforth em-ployed to solve the initial value problem (i.v.p.) corresponding to decayingdata for a number of important nonlinear evolution equations appearing inphysics, see [6–8].

KP is the compatibility of the linear spectral equations Lψ(x, y) =Mψ(x, y) = 0, where the operator L is either the nonstationary Schrodinger(NS) equation when ε2 = −1 or the heat operator with ε2 = 1 defined as

Lψ ≡ [ε∂y + ∂xx + u(x, y)]ψ(x, y) , (1.2)

and

Mψ ≡ (∂t + 4∂xxx + 6u∂x + 3ux − 3ε(∂y∂−1x u) + h(k))ψ = 0 (1.3)

where h(k) is an arbitrary function [9].The solution of the Cauchy problem with decaying data for equations

(1.1) is connected to the solution of the spectral problem (1.2) as we explainbelow. The IST for KPII involves using a ∂ problem [4]. The relevant ideasare developed in Section 2. Another feature of KPII is the appearance ofline solitons: real and localized solutions that decay exponentially at infinityeverywhere except along a line. From a dynamical perspective these are alsoasymptotically free objects moving with uniform velocities in which theonly effect of interaction is a certain translation, which is usually termeda “phase shift”. These configurations seem to be generic and stable underKPII evolution. We note that although line solitons were known as early as1976 [10], they are not recovered by IST formulations in [4] as they are offthe class studied. To explain the appearance of line solitons of KPII, newideas are necessary that we discuss in Section 3.

In Section 4 we consider initial value problems (i.v.p.) for KPI. The ISTinvolves using a nonlocal Riemann problem [3]. Despite the fact that KPIalso has line soliton solutions, they are unstable (cf. [1, 2, 7]). However,it supports multi-lump solutions; these are localized configurations thatdecay rationally everywhere which possess simple dynamics. A class of suchsolutions is constructed in [11]. This class corresponds to asymptoticallyfree objects that move with uniform velocities and decay rationally (wecall this an N -lump solution if the result consists of N of these objects).The dynamics is characterized by an elastic interaction between the lumps,which is the typical behavior for localized solitonic solutions of integrableequations. From a spectral perspective, the N -lump solution is associatedwith wave functions that have simple poles in the spectral variable [3].Section 5 develops the relevant ideas.

Localized real valued solutions with rational decay but exhibiting non-trivial asymptotic dynamics have been found for KPI and several otherintegrable equations [12]–[15]. The connection with the discrete spectrum

SOLUTIONS OF THE KP EQUATION 3

of the relevant spectral problem has been discussed in detail for the case ofthe NS and KPI equations [16]–[18]. Spectrally, these potentials correspondto wave functions that have higher order poles and have an associated wind-ing number Q, an integer greater than one; while standard noninteractinglumps correspond to Q = 1 (see also [19, 20] regarding direct methods usedto examine the properties of these NS and KPI solutions, and [21, 22] inconnection with the Davey–Stewartson eqs.). We develop the relevant ideasin Section 5.

Section 6 discusses the rigorous aspects of the direct and inverse prob-lems associated with both KPI and KPII. Conditions for existence anduniqueness are given in terms of some natural functional norms. Finally inSection 7 we consider the role of the constraints with regard to the initialvalue problem. We find that, with data not satisfying these constraints,a KP evolution is possible but the corresponding solutions decay weakly,irrespective of the decay of the initial data.

2. The Cauchy problem for KPII with decaying potentials

As mentioned above, the KPII equation

(ut + uxxx + 6uux)x + 3uyy = 0 (2.1)

is associated to the the heat operator with a potential defined as

Lψ ≡ [−∂y + ∂xx + u(x, y)]ψ(x, y) = 0 (2.2)

and

Mψ ≡ (∂t + 4∂xxx + 6u∂x + 3ux + 3(∂y∂−1x u) + h(k))ψ = 0 . (2.3)

We assume that the initial data u(x, y, t = 0) is real, nonsingular and hassuitable decay as x2 + y2 →∞ and that

∫dxu(x, y, t = 0) = 0. Consider a

new function μ(x, y, k) via ψ(x, y) = μ(x, y, k) exp(ikx− k2y) that includesthe complex spectral parameter k ≡ kR + ikI so μ satisfies

(−∂y + ∂xx + 2ik∂x)μ(x, y, k) = −u(x, y)μ(x, y, k) . (2.4)

We consider an eigenfunction of (2.4) that is bounded and tends to 1 asx2 + y2 → ∞. Under this proviso, (2.4) can be converted into the integralequation

μ(x, y, k) = 1 +

∫dx′dy′G(x− x′, y − y′, k)uμ(x′, y′, k) . (2.5)

G(x, y, k) is an appropriate Green’s function for (2.4), i.e., it solves

(−∂y + ∂xx + 2ik∂x)G(x, y, k) = −δ(x)δ(y) (2.6)


selected by requiring that the eigenfunction μ(k) is bounded. By Fouriertransformation we find that

G(x, y, k) =1

(2π)2

∫ ∞

−∞dp

∫ ∞

−∞dq exp [i(px+ iqy)] /(p2 + 2pk + iq) . (2.7)

Note that the integrand has integrable singularities at the points p = q = 0and p = −2kR, q = 4kRkI ; hence the Green’s function is well defined. Notealso that they depend separately on both real and imaginary parts of thespectral parameter. This property also holds for the Green’s function, whichis therefore analytic nowhere. Actually, upon integration one obtains

G(x, y, k) =1

2π

∫dp eipx−y(p

2+2kp)[θ(y)(χA+θ(kR) + χA−θ(−kR))

− θ(−y)(χAc+θ(kR) + χAc

−θ(−kR)

], (2.8)

where we define the sets

A+ ≡ (−∞,−2kR) ∪ (0,∞); A− ≡ (−∞, 0) ∪ (−2kR,∞), (2.9)

Ac± stands for:

Ac+ ≡ IR−A+ = [−2kR, 0], Ac− ≡ IR−A− = [0,−2kR],

and the indicator function χA of the set A ⊂ IR is defined as

χA±(p) =

{1, p ∈ A±,0, p /∈ A± .

Note that, as a function of k, G(x, y, k) is not holomorphic anywhere.Besides, it has different representations according to whether kR > 0, orkR < 0. However, it is easily seen to be continuous across kR = 0 (see below(2.11)).

We next list a few useful properties of the Greens function relevant inthis connection.

i) The first one is the departure from holomorphicity measured by ∂G∂k(k).

By direct calculation on formula (2.8) and using ∂/∂k = 12(∂/∂kR +

i∂/∂kI) we find that

∂G

∂k(x, y, k) = λeiqy+ipx . (2.10)

where λ ≡ − signkR2π ; p = −2kR, q ≡ 4kRkI .

ii) G is continuous across kR = 0 and

G(x, y, k = ikI) =1√4πy

ekIx−k2Iy−x2

4y θ(y) . (2.11)


iii) G satisfies the symmetry relationship

G(x, y, k) = eiqy+ipxG(x, y,−k) . (2.12)

iv)lim|k|→∞

G(x, y, k) = 0 . (2.13)

The integral equation (2.5) is of Fredholm type. Existence and uniquenessof its solutions can be studied by iteration [4, 23]. We prove in chapter 6 thatexistence and uniqueness is guaranteed when the initial data u0 ≡ u(x, y, 0)has L1 and L2 norms that satisfy ‖u‖1‖u‖22 < 0.26, or more concretely

54‖u‖1‖u‖22C2 < 1 where C ≡ 1

(2π)2

∫ 2π

0

dϕ√| sinϕ| .

See (6.11) below.If the latter condition is not satisfied, solutions φ to the homogeneous

version of (2.5) might exist at some points kα (the eigenvalues). This wouldbe similar to what happens for KPI (see Section 5); however the existenceof real, nonsingular, localized modes has never been proven. In this regardwe note that corresponding to kR = 0 it is easy to establish that no suchmodes exist; indeed, in this case, (2.11) shows that the integral equation(2.5) is of Volterra type, precluding the existence of eigenvalues.

The next step of the IST transformation is the so called inverse prob-lem. It involves reconstructing the eigenfunction μ(k) ≡ μ(x, y, k) fromappropriate data. As has already been commented, μ(k) generically is notholomorphic anywhere. We evaluate its departure from holomorphicity byusing the ∂ derivative of the Greens function. By direct calculation, andusing (2.10), we find the following integral equation for ∂μ

∂k:

∂μ

∂k(x, y, k) = F (k)eiqy+ipx +

∫dx′dy′G(x− x′, y − y′, k)u∂μ

∂k(x′, y′, k) ,

(2.14)where

F (k) ≡ λ∫dxdye−iqy−ipxuμ(x, y, k) (2.15)

defines the bidimensional scattering data of the problem. Using (2.12) itthen follows that μ and its ∂ derivative are related as follows:

∂μ

∂k(x, y, k) = eiqy+ipxF (k)μ(x, y,−k) . (2.16)

Once the departure from holomorphicity of the function μ is evaluated,the inverse problem is formulated using the normalization

μ(k) = 1 +μ1k+O(1/k2), |k| → ∞, where 2ik∂xμ1 = −u , (2.17)


which follows from (2.5), (2.13), and the generalized Cauchy formula

μ(k) =1

2πi

∫D∞

μ(z)

z − kdz +1

2πi

∫D

∂μ∂z

z − kdz ∧ dz , (2.18)

where D is a large disk in the complex plane and D∞ its boundary. Interms of the scattering data described above, we have

μ(x, y, k) = 1 +1

2πi

∫C|F (z)μ(x, y,−z)e

2izR(2zIy−x)

z − k dz ∧ dz . (2.19)

Eq.(2.17) implies that the potential is obtained from

u =1

π

∂

∂x

∫C|F (z)μ(x, y,−z)e2izR(2zIy−x)dz ∧ dz . (2.20)

We next determine the temporal evolution of the scattering data. To thisend, recall that ψ(x, y, k) must also solve (2.3). Requiring the normalizationψ(x, y) → exp(ikx − k2y) as x2 + y2 → ∞ to hold for all time implies,using the fact that the potentials vanish at ∞, that h(k) = 4ik3. Note alsothat ∂kψ also solves M∂kψ = 0, since M does not depend on k: ∂kMψ =M∂kψ = 0. Inserting (2.16) in the latter equation one obtains that the dataF (k) must evolve in time via

∂

∂tF (k, t) = −4iF (k, t)(k3 + k3) , (2.21)

i.e.,

F (k, t) = F (k, 0)e−4i(k3+k3)t . (2.22)

Hence the solution u(x, y, t) of the KPII equation is obtained from eqs.(2.19), (2.20) after the time dependence of F is inserted. In Section 6 rigor-ous results associated with the IST analysis (direct and inverse scattering)for the KPII eq. are described.

3. Initial value problem for KPII with potentials supported alonga line

Natural interesting solutions of KPII, which seem to be generic and stableunder KPII evolution, are the line solitons which are real and localizedsolutions that decay exponentially at infinity everywhere except along aline. We note that they are not included in the IST formulation of thelatter section as they are off the class studied. The Cauchy problem in theplane for eq. (2.1) corresponding to physical data which is real, nonsingularand decays at infinity everywhere except along a line involves new ideas that


have been developed recently [24]. Here we summarize our work regardingthis situation. More precisely, we consider general initial data u(x, y, t = 0)that is real, nonsingular and decays at infinity everywhere except alonga line L on the plane: L = {(x, y)|x − vy = x0} (here x0, v are theparameters defining the line) i.e.: limx,y→∞ u(x, y) = 0 for (x, y) /∈ L and if(x, y) ∈ L, limx,y→∞ u(x, y) = u∞(x0). Here u∞(x0) is an arbitrary givenfunction that we assume is smooth and decaying: limx0→∞ u∞(x0) = 0.It may contain, in particular, a superposition of solitons. The potential isdecomposed into an asymptotic nondecaying component along a line and adecaying contribution: u(x, y) = u∞(x− vy)+U(x, y). (We note that otherinteresting studies regarding the inverse scattering associated with the heatoperator with one soliton potential plus a decaying background have beenpublished (cf. [25]- [30]).

Important features of these results are:

i) The inverse problem is a combination of a ∂ contribution due to U(x, y)plus a Riemann–Hilbert (RH) contribution arising from the reflectioncoefficient obtained from u∞(x) plus pole contributions that are dueto the discrete eigenvalues associated with u∞(x). Generically, thespectrum is a combination of these three terms.

ii) If u∞ vanishes the formalism reduces to the ∂ solution of the lastsection.

iii) The Green’s function is bounded in the spectral plane and is contin-uous away from the discrete eigenvalues; in particular, it has no linediscontinuities.

iv) Completeness relations for the time independent Schrodinger operatorare important in establishing the properties of the Green’s function.

3.1. DIRECT AND INVERSE PROBLEMS

Given eq. (2.2), consider a new function μ(x, y, k) via

ψ(x, y, k) = μ(x, y, k) exp

[− ik(x− vy)− v

2x+

(v24− k2

)y

](3.1)

that includes the spectral parameter k, and new coordinates x ≡ x − vy,y ≡ y. In the new frame, and after dropping tildes, we find that the spectralproblem (2.2) associated to KPII is conveniently transformed into

Lμ(x, y, k) = −U(x, y)μ(x, y, k);

L ≡ −∂y + ∂xx − 2ik∂x + u∞(x) .(3.2)


Following [24], we next discuss how to construct a Green’s function

G(x, x′, y, k) for the operator L:

(−∂y + ∂xx − 2ik∂x + u∞(x))G(x, x′, y, k) = −δ(x− x′)δ(y) . (3.3)

Define solutions φ±(x, k), ψ±(x, k) to the equations

(∂xx + k2 + u∞(x))φ±(x, k) = 0;

(∂xx + k2 + u∞(x))ψ±(x, k) = 0

(3.4)

by requiring the conditions

φ±(x, k)x→−∞= e∓ikx ; ψ±(x, k)

x→∞= e±ikx . (3.5)

(Note that ψ±(x, k) is not directly related to the wave function ψ(x, y, k).)If u∞(x) satisfies

∫(1 + |x|2)|u∞(x)|dx < ∞, the former functions exist

and are analytic functions of k ≡ kR + ikI on C| ± (the upper/lower half kplanes), having limits on the boundary {kI = 0} (cf. [31, 32]). These limitssatisfy the following relation:

φ+(x, k) = a(k)ψ−(x, k) + b(k)ψ+(x, k), k ∈ IR , (3.6)

for certain functions a(k), b(k). The function a(k) can be proven to beanalytic in k on the upper half plane, having a denumerable set of (simple)zeros {kj ≡ iχj, χj ∈ IR+}j=1...N . If ψj(x) ≡ ψ+(x, kj) then φ+(x, kj) =βjψj(x) for some complex constant βj . We call a+(k) ≡ a(k).

Define next

Gc(x, x′, y, k) = Gc+(x, x′, y, k)θ(kI) +Gc−(x, x′, y, k)θ(−kI) , (3.7)

where

Gc± =1

2π

∫C′±

dp eik(x−x′)−y(p2+2kp)g±(x, x′, p+ k)

× [θ(y)(χA+θ(kR) + χA−θ(−kR))− θ(−y)(χAc

+θ(kR) + χAc

−θ(−kR))] , (3.8)

g+(x, x′, k) ≡ φ+(x, k)ψ+(x

′, k)a+(k)

, kI > 0 (3.9)

g−(x, x′, k) ≡ −φ−(x′, k)ψ−(x, k)

a−(k), kI < 0 , (3.10)

and A± are defined in (2.9). C ′± is the contour along the real axis witha small semicircular indentation below (−)/above (+) the point p = −kR(corresponding to the zeros of a(p+ k) when p+ k = iχj). Let also

Gd(x, x′, y, k) = i

∑|kj|≥|kI |

Cjeik(x−x′)+(k2+χ2

j)yψj(x)ψj(x

′)θ(−y) (3.11)


(the discrete part of the Green’s function). Finally we consider

G(x, x′, y, k) = Gc(x, x′, y, k) +Gd(x, x′, y, k) . (3.12)

It can be verified that the Green’s function satisfies (3.3). Note that in thecase that u∞ = 0

φ±(x, k) = e∓ikx; ψ±(x, k) = e±ikx, a(k) = 1, {kj} = ∅ ,and then Gd = 0, G+ = G−, and the Green’s function reduces (upon lettingk → −k) to that of the latter section. We also note that G is bounded andcontinuous everywhere except at the points k = iχj, where it has limitsfrom both sides kI → χ+j , kI → χ−j but it has a jump.

We consider a solution μ(x, y, k) to (3.2) via

μ(x, y, k) = h(x, k) +

∫dx′dy′G(x, x′, y − y′, k)Uμ(x′, y′, k) , (3.13)

where

μ =

{μ+(x, k), kI > 0μ−(x, k), kI < 0 ;

(3.14)

h(x, k) =

{eikx φ+a (x, k), kI > 0

eikxψ−(x, k), kI < 0 .(3.15)

Operating with ∂∂kon equations (3.8), (3.13) we find the following integral

equation for ∂μ±∂k

corresponding to k ∈ C| ±:

∂μ±∂k

(k) = πN∑j=1

Cje−χjxψj(x)δ(kR)δ(kI − χj)

+ ei(qy−px)F±(k)h±(x,−k)

+

∫dx′dy′G±(x, x′, y − y′, k)U

∂μ±(k)∂k

, (3.16)

where λ ≡ − signkR2π , q ≡ 4kRkI , p = −2kR. The functions

F+(k) ≡ λ

∫dxdye−ikx−iqyUμ+(x, y, k)ψ+(x,−k)

F−(k) ≡ −λ

a−(−k)

∫dxdy e−ikx−iqyUμ(x, y, k)φ−(x,−k) .

(3.17)

define the bidimensional scattering data of the problem. It then follows that

∂μ±∂k

= πN∑j=1

Cje−2χjxμ−(−iχj)δ(kR)δ(kI − χj)

+ ei(qy−px)F±(k)μ±(−k) . (3.18)


The function μ(x, y, k) has a jump on the real axis that satisfies

[μ+ − μ−](x, y, k) = ρ(k)e2ikx μ−(x, y,−k) . (3.19)

Therefore, the global function μ(k) satisfies

∂μ

∂k= π

N∑1

Cje−2χjxμ−(−iχj)δ(kR)δ(kI − χj) (3.20)

+ ei(qy−px)∑±F±(k)μ±(−k)θ(±kI)−

ρ(k)

2ie2ikxμ−(−k)δ(kI ) . (3.21)

Once the departure from holomorphicity of the function μ is evaluated,the inverse problem follows from the generalized Cauchy formula (2.18)

μ(k) = 1 +N∑1

Cje−2χjxμ−(−iχj)

k − iχj

+1

2πi

∑±

∫C±F±(z)μ±(−z)

ei(qy+2zRx)

z − k dz ∧ dz

+1

2πi

∫ ∞

−∞dzR

ρ(zR)e2izxμ−(−zR)zR − k

. (3.22)

From (3.22) we obtain a coupled system of integral equations after appro-priately evaluating the terms for k ∈ C| ±. Then, the potential is obtainedfrom

u =∂

∂x

[2i

N∑1

Cje−2χjxμ−(−iχj)

− 1

π

∑±

∫C| ±F±(z)μ±(−z)ei(qy+2zRx)dz ∧ dz (3.23)

− 1

π

∫ ∞

−∞dzRρ(zR)e

2izRxμ−(−zR)], (3.24)

and, after restoring the initial coordinates, u is given by (3.24) letting x→x− vy.

The fact that ν+ has simple poles at k = iχj , j = 1, 2, . . . , n, suggeststhat the inverse problem (3.22) has a double pole, viz. a nonintegrablesingularity. However, the singularity is less severe. Indeed letting μ+ =a(k)μ1 implies that F+ = (λF1(k))/a(k), where

F1(k) ≡∫dxdye−iqyUμ1(x, y, k)ψ+(x,−k) .


Evaluating this at k = ikI �= iχj , using eq. (3.2) and employing the bound-ary conditions one can prove that limkI→κj F1(kR = 0, kI) = 0. Actually,the singularity is at most a simple pole; one can prove the following:

limkI→κj

∂F1∂k

(kR = 0, kI) = limkI→κj

∂F1∂k

(kR = 0, kI) = 0 .

3.2. TEMPORAL EVOLUTION AND LINE SOLITONS

The temporal evolution is obtained using again (2.3). Corresponding to ourboundary conditions we have, for all time, the normalization

ψ(x, y)→ exp[− ik(x− vy)− v

2x+

(v24− k2

)y], x2+ y2 →∞ . (3.25)

Hence we take h(k) = 4(v2 + ik)3. This implies that

∂

∂tF±(k, t) = F±(k, t)(h(−k)− h(k)) , (3.26)

and hence

F±(k, t) = F±(k, 0)e4[(v2−ik)3−( v

2+ik)3]t . (3.27)

The rest of the scattering data are found to satisfy

∂

∂tρ(k, t) = 4

[(v2− ik

)3−(v2+ ik

)3]ρ (3.28)

∂

∂tCj(t) = 4

[(v2+ χj

)3−(v2− χj

)3]Cj(t) , (3.29)

so that

ρ(k, t) = ρ(k, 0)e4[(v2−ik)3−( v

2+ik)3]t ;

Cj(t) = Cj(0)e4[( v

2+χj)3−( v2−χj)3]t .

(3.30)

Hence the solution u(x, y, t) of the KPII equation is obtained from eqs.(3.19, 3.20) after the time dependence of F±, ρ, Cj is inserted.

When the continuous scattering data are all zero: F+(k) = F−(k) =ρ(k) = 0, the equations to recover the eigenfunction μ in modified coordi-nates (reminding the reader that x = x− vy) are

ζl = Cle−χlx −

N∑1

Cle−(χj+χl)x

i(χl + χj)ζj , (3.31)


where we call Cjμ(−iχj)e−χj x ≡ ζj . Solving this system in the same way asfor KdV, restoring the original coordinates and introducing the temporalevolution we obtain that the solution is given in terms of this data by

u(x, y, t) = 2d2

dx2log detF (x− vy, t) , (3.32)

where the N ×N matrix (Flj)N×N is defined by

Flj(x, t) = δlj − iCle−(χj+χl)x+8χj(χ2j+3 v2

4)t

χj + χl. (3.33)

The solutions are line solitons, all of them moving along the same direction,and with properties similar to that of the standard soliton. In particularwith N = 1, χ1 ≡ χ

u(x, y, t) = 2χ2sech2χ[(x− vy)− (4χ2 + 3v2)t− x0] , (3.34)

where x0 =12χ log (

C1(0)2iχ ).

4. KPI equation with decaying initial data

We now consider the i.v.p. for the KPI equation given by (1.1) with ε = icorresponding to decaying initial data u(x, y, 0). The relevant operator isnow the nonstationary Schrodinger operator (1.2). We follow [3]. As above,

we define a new eigenfunction μ(x, y, k) = ψ(x, y)e−ikx+ik2y, that solves

0 = (i∂y + ∂xx + 2ik∂x + u)μ(x, y, k) ≡ Lμ . (4.1)

We consider solutions μ to (4.1) that correspond to regular and decayingpotentials u(x, y) tending to 1 as x2 + y2 →∞. Eq. (4.1) can be convertedinto the integral equation

Gμ(x, y, k) = 1 , (4.2)

where the operator G(k) is as follows

Gf(x, y) = f(x, y)−∫ ∞

−∞dx′

∫ ∞

−∞dy′G(x− x′, y − y′, k)(uf)(x′, y′) ,

G(x, y, k) =1

(2π)2

∫ ∞

−∞dp

∫ ∞

−∞dq exp [i(px+ iqy)] /((p2 + 2pk + q) .

G(x, y, k) is an appropriate Green’s function selected by requiring that theeigenfunction μ(k) admits an analytic extension to the complex k plane. Wedenote the analytic extensions to the upper (+)/lower (−) half planes by


G±(k). Note that the integrand has an integrable singularity at the pointp = q = 0 if kI �= 0, while there is an entire parabola of singularities atthe points (p,−p2 − 2kRp) if kI = 0. It follows that if kI �= 0 the Green’sfunction is well defined. Upon integration one obtains

G±(x, y, k) = −1

2πi

∫ ∞

−∞dp exp

[i(px− (p2 + 2kp)y)

]× (θ(y)θ(∓p)− θ(−y)θ(±p)) . (4.3)

A few useful properties of this function are the following:

G+ −G− =1

2πi

∫ ∞

−∞sign(p) exp

[i(px− (p2 + 2kp)y)

], (4.4)

Im k = 0 ;

∂G(x, y, k)

∂k= −i(x− 2ky)G(x, y, k) − 1

2πisign(Im k), (4.5)

Im k �= 0 ;

G(x, y, k) = G(−x,−y, k) , (4.6)

where the bar denotes complex conjugation.Likewise, the analytic extensions to the upper (+)/lower (−) half planes

of μ(k), that we denote μ±, satisfy

μ±(x, y, k) = 1−∫ ∞

−∞dx′

∫ ∞

−∞dy′G(x− x′, y − y′, k)(uμ±)(x′, y′, k). (4.7)

Existence of solutions to this equation is discussed in Section 6.In order to determine the RH problem governing the inverse problem

we next evaluate the jump Δ(k) ≡ Δ(x, y, k) ≡ μ+(k) − μ−(k) that μ(k)has across the real axis. Note that Δ solves

Δ =

∫ ∞

−∞dx′

∫ ∞

−∞dy′(G+ −G−)(x− x′, y − y′, k)(uμ+)(x′, y′, k)

+

∫ ∞

−∞dx′

∫ ∞

−∞dy′G−(x− x′, y − y′, k)(uΔ)(x′, y′) . (4.8)

Hence using (4.4) we have

Δ =

∫ ∞

−∞T+(k, l)e

β(x,y,k,l)dl

+

∫ ∞

−∞dx′

∫ ∞

−∞dy′G−(x− x′, y − y′, k)(uΔ)(x′, y′) , (4.9)

where

β(x, y, k, l) ≡ ei[(l−k)x−(l2−k2)y]

T+(k, l) =sign(l − k)

2πi

∫ ∞

−∞dx

∫ ∞

−∞dy(uμ+)(x, y, k)e

β(x,y,k,l) .


It follows that

Δ(x, y, k) ≡ μ+(k)− μ−(k) =∫ ∞

−∞T+(k, l)N(x, y, k, l) dl , (4.10)

where N(x, y, k, l), k, l ∈ IR solves

N(x, y, k, l) = eβ(x,y,k,l)

+

∫ ∞

−∞dx′

∫ ∞

−∞dy′G−(x− x′, y − y′, k)(uN)(x′, y′, k, l) . (4.11)

Note that the eigenfunctions N(x, y, k, l), μ−(x, y, k) are related by

μ−(x, y, k) = N(x, y, k, k) . (4.12)

We next establish the relationship between N(x, y, k, l) and μ−(x, y, k). De-fine f(x, y, k) = f(x, y, k)ei(kx−k2y); then note that N(x, y, k, l) =

solves

N(x, y, k, l) = ei(lx−l2y)

+

∫ ∞

−∞dx′

∫ ∞

−∞dy′G−(x− x′, y − y′, k)(uN )(x′, y′, k, l) . (4.13)

Hence e−i(kx−k2y)∂k N(x, y, k, l) ≡ π(x, y, k, l) solvesπ(x, y, k, l) = F (k, l)

+

∫ ∞

−∞dx′

∫ ∞

−∞dy′G−(x− x′, y − y′, k)(uπ)(x′, y′, k, l) , (4.14)

which implies that π(x, y, k, l) = F (k, l)μ−(x, y, k). Here we define

F (k, l) =1

2πi

∫ ∞

−∞dx

∫ ∞

−∞dy(uN)(x, y, k, l) . (4.15)

By integration (taking (4.12) into account) one obtains

N(x, y, k, l) = μ−(x, y, l)eβ(x,y,k,l)

+

∫ k

lF (p, l)μ−(x, y, p)eβ(x,y,k,p) p , (4.16)

which is the desired relationship. Eq. (4.16) expresses N(x, y, k, l) in termsof μ−(x, y, k) and of the data F (p, l). HenceN(x, y, k, l) must be determineddirectly from (4.11), and then F (k, l) follows from (4.15).

It turns out that T+(k, l) and N(x, y, k, l) do not show directly in thefinal representation of the jump. Only F (k, l) and μ−(k) do so. Indeed,using (4.10) and (4.16) we obtain (see [3])

μ+(x, y, k) − μ−(x, y, k) =∫ ∞

−∞dlf(k, l)(uμ−)(x, y, l)eβ(x,y,k,l) , (4.17)

N(x, y, k, l)ei(kx−k2)


and f(k, l) ≡ sign(k − l)F (k, l).Alternatively, one can obtain the data F (k, l) from the following integral

equation (use (5.2.29) of ref. [8]):

F (k, l) = S(k, l) +

∫ k

lS(k, p)F (p, l) dp , (4.18)

where we define

S(k, l) =1

2πi

∫ ∞

−∞dx

∫ ∞

−∞dy (uμ−)(x, y, l)eβ(x,y,k,l) . (4.19)

The determination of the jump is thereby reduced to solving the linearintegral eqs. (4.7), (4.17) and either (4.11) or (4.18).

The inverse problem equation and the potential are obtained using theformulae (2.17), (2.18), (4.17). After some transformations one obtains

μ(x, y, k) = 1 +1

2πi

∫ ∞

−∞

∫ ∞

−∞f(p, l)eβ(x,y,p,l)μ−(x, y, l)

dp dl

p − k + i0 (4.20)

u(x, y) = − 1π

∂

∂x

∫ ∞

−∞

∫ ∞

−∞f(p, l)eβ(x,y,p,l)μ−(x, y, l) dp dl . (4.21)

Solutions to KPI are obtained after inserting the relevant time dependence,which in this case reads

F (k, l, t) = F (k, l, 0) exp 4i(l3 − k3)t . (4.22)

Notes

1. We assume that μ± exist and have analytic extensions to the upper(+)/lower (−) half planes, respectively, and that they do not havesingularities. (In Section 6 we discuss conditions that guarantee theseproperties. The addition of pole singularities is discussed in Section 5).

2. Formula (4.20) for the inverse problem was first derived in [5], althoughno connection to initial value problems was drawn; i.e., it remained toestablish how the inverse problem and the scattering data follow fromthe direct side and to connect to the i.v.p.. This is accomplished from(4.11) and (4.15) (or from (4.18)).

5. Discrete modes for the KPI equation with decaying initialdata


5.1. CHARACTERIZATION OF THE DISCRETE SPECTRUM OF THE KPIEQUATION

In the previous section we considered real, nonsingular potentials that de-cay at infinity, which correspond to solutions μ to (1.2) that satisfy (4.2).Existence and uniqueness of the Fredholm-like eq. (4.2) is obtained (see[33, 34]) when the initial data u(x, y, 0) satisfy the small norm condition∫ ∫

|u(p, y, 0)|dydp < 1 , u(p, y, 0) ≡∫dxe−ipxu(x, y, 0) . (5.1)

Other conditions directly in terms of norms on the physical space arederived in Section 6 below.

When condition (5.1) does not hold, μ(k) may have singularities that weassume to be poles. The case when both the pole’s multiplicity and a certaininteger (referred to as the charge or index, see below (5.10)) are unity, aswell as the necessary modification of (4.20), was discussed in [3]. In thissection we discuss the situation when higher-order pole multiplicities and/orhigher charges, are allowed. It seems that the notion of index is absentfor one-dimensional problems and only appears in multidimensions; seealso [45] regarding multiple poles for the Dirac operator in one dimension.Around any pole k1 we have the Laurent expansion

μ(k) =∞∑r=0

ρrr!(k − k1)r +

M∑r=1

Ψr(k − k1)r

= ν(k) +Φ

(k − k1)+

M∑r=2

Ψr(k − k1)r

,

(5.2)where Ψ1 ≡ Φ, ν(k1) = ρ0 and

ν(k) =∞∑r=0

ρrr!(k − k1)r (5.3)

is the regular part of μ(k) at k1, while μ(k) − ν(k) is the singular part ofμ(k) at k1. The next result establishes the connection between the polestructure of μ(k) and the discrete spectrum.

5.1.1. Result IAssume an eigenfunction μ(k) with a meromorphic representation exists:

μ(k) = 1 +n∑α=1

{Φα

k − kα+Mα∑r=2

Ψr,α(k − kα)r

}, (5.4)

where Φα,Ψr,α depend only on x, y. Note that

ν(kα) = 1 +n∑

β=1,β �=α

{Φβ

kα − kβ+

Mβ∑r=2

Ψr,β(kα − kβ)r

}. (5.5)


At any point k1 the (negative) Laurent coefficients of μ(k) around k1:

{ν(k1),Φ1,Ψr,1}, r = 2, . . . ,M, M ≥ 2 ,

satisfy the integral equations

{GΨM,1}k=k1 ={GΨM−1,1 +

∂G∂kΨM,1

}k=k1

= · · ·

={GΦ1 +

∂G∂kΨ2,1 + · · ·+

1

(M − 1)!∂M−1G∂kM−1

ΨM,1}k=k1

= 0 ,

{Gν + ∂G

∂kΦ1 +

1

2

∂2G∂k2

Ψ2,1 + · · ·+1

M !

∂MG∂kM

ΨM,1}k=k1

= 1. (5.6)

Proof.Take the limit k → k1 in eq. (4.2) and set the coefficients of

1/(k − k1)j , j = 0, 1, . . . ,M ,

equal to zero.

If (5.1) is not met, then eigenfunctions or solutions ω(x, y) to the ho-mogeneous version

Gk1ω ≡ ω(x, y)

−∫dx′

∫dy′G(x− x′, y − y′, k1)u(x′, y′)ω(x′, y′) = 0 (5.7)

may exist at some points k1 (the eigenvalues), where we denote Gk1 ≡ G(k =k1). These are called homogeneous solutions of the operator Gk1 . The span(vector space) of all such functions is denoted KerGk1 . They define thediscrete spectrum of the nonstationary Schrodinger operator (4.2), and areresponsible for the appearance of the lumps, i.e., real, nonsingular potentialsthat decay rationally at infinity. The relevant theory for standard lumps wasobtained in [3]. The general situation is however much more elaborated, andwas developed in the articles [16]–[18].

We next review the relevant ideas (the detailed proofs of the main resultcan be found in ref. [17]). Call ρ(x, y) the homogeneous adjoint solution,which satisfies

G†k1ρ ≡ ρ(x, y)−∫dx′

∫dy′G(x′ − x, y′ − y, k1)u(x, y)ρ(x′, y′) = 0 . (5.8)

We reiterate that certain properties of these operators, e.g. compactnessand decay of their solutions, is assumed in this discussion.


5.1.2. Result IIAt any point k1 the number of linearly independent homogeneous solutions

to Gk1ω1 = 0, G†k1ρ = 0 and Gk1ω1 = 0 is the same. Result I implies that thepoles of μ(k) are eigenvalues, i.e., they are points of the discrete spectrumof (4.2), while ΨM,α are eigenfunctions of (4.2) at the eigenvalue kα. ByResult II we have that the discrete spectrum is an even dimensional set{kj , kj}j=1,...,N . As a consequence, pure meromorphic functions μ have thestructure (5.4), where the poles occur in pairs (kj , kj+N ) ≡ (kj , kj), andhence without loss of generality their position can be arranged such thatIm (kj) > 0; j = 1, 2, . . . , N .

Suppose μ corresponds to the discrete spectrum of the Schrodinger oper-ator and has 2N poles {kj , kj}j=1,...,N with multiplicities {Mj ,Mj}j=1,...,N ,and μ→ 1 as k →∞. Such an eigenfunction is given by

μ(k) = 1 +N∑j=1

{Φj

k − kj+

Φjk − kj

+

Mj∑r=2

Ψr,j(k − kj)r

+

Mj∑r=2

Ψr,j(k − kj)r

}, (5.9)

where we have used kj+N = kj in (5.4).Fundamental objects in our development are certain quantities that we

call indices (winding numbers) or charges. At any pole kα, α = 1, . . . , 2Nwe define the index of the pole as

Qα ≡1

2πisign(Im kα)

∫ ∫uΦα dxdy , (5.10)

where Φα is the residue of μ at the pole. We also introduce the secondaryindices

Qr,α ≡1

2πisign(Im kα)

∫ ∫uΨr,α dxdy, 2 ≤ r ≤Mα . (5.11)

Note: to stress the importance of the charges Q1,α, we drop the subscriptand write Qα ≡ Q1,α.

5.1.3. Result IIILet μ(k) be given by eq. (5.9) with discrete spectrum {kj , kj}Nj=1. Thenthe indices Qα have the following properties:

i)Qα = Qα and Qr,j = Qr,j . (5.12)

ii) Eigenfunctions corresponding to eigenvalues k1, k2 with k1 �= k2 areorthogonal with respect to the inner product

〈f, g〉 ≡ 1

π

∫fx g dxdy (5.13)

If k1 = k2 their scalar product equals the “charge” Q.


iii) The charge Q can take only positive integers values.

iv) Q has a topological interpretation as

Q = winding number H (5.14)

where H is related to the residue Φ of the eigenfunction μ satisfyingeq. (4.2) by

Φ = i∂

∂flogH +O(1/r2), r2 ≡ x2 + y2 →∞ , (5.15)

where f ≡ x− 2k1y, and the derivative is taken along a closed contourat infinity Γ∞ surrounding the origin once. In fact we find H = fQ +O(1/r) as r→∞.

v) Solutions of the nonstationary Schrodinger operator (eq. (1.2),(4.2))can have higher-order poles.

vi) Characterization of the class of localized potentials requires the fol-lowing information: the position and multiplicity Mj of the poles atkj , the associated indices Qj and the dimension rj ≥ 1 of the nullspace of the corresponding eigenfunctions. In this regard we note thatwe have found that the Fredholm conditions yield at k0 the followingrelationship between these numbers to hold at any eigenvalue

M0 +M0 + r0 − 2 = Q0 , (5.16)

and hence in particular M0 +M0 − 1 ≤ Q0, where M0,M0 are theorder of the poles of the eigenfunction at k0, k0 resp. We have in factfound from direct methods for all these solutions that the following“two sided” inequality (see ref. [20]) holds:

M0 +M0 − 1 ≤ Q0 ≤M0M0 . (5.17)

vii) Meromorphic wave functions correspond to real, nonsingular and de-

caying potentials of the form u(x, y) = 2 ∂2

∂x2logF (x, y, γ, δ, . . .) where

F is a certain polynomial and γ, δ, . . . constants. They also correspondto localized, real, nonsingular, decaying solutions of the KPI equationwhen time dependence is inserted. The dynamical evolution of theseconfigurations is nontrivial, as opposed to the situation for standardlumps or solitons.

Remark.


1. Standard lumps potentials of KPI correspond to having simple polesfor which the index Q = 1 [3]. As we have just seen, in the generalsituation poles of any multiplicity are allowed, as well as higher ordervalues of the index.

2. The inner product (5.13) had also appeared in [35].

5.2. EQUATIONS DETERMINING THE EIGENFUNCTION

In this section we discuss a method for determining equations which fix theeigenfunctions μ(k) satisfying eq. (4.2) with certain assumed analytic struc-ture. Once these equations have been determined, we show in Section 5.3how they can be used to solve for particular cases of eigenfunctions andrelated reflectionless multipole potentials which are real, nonsingular anddecaying at infinity. A linear relationship is obtained.

5.2.1. Simple polesWe begin by analyzing simple pole solutions of eq. (4.2), i.e., special cases ofeq. (5.4). Assume that the solution to eq. (4.2) has the following structurearound k = k1:

μ(k) = ν(k) +Φ

k − k1, (5.18)

where ν(k) is regular at k1 and ν(k) −→k→∞

1. Define recursively

Jn(f, y), n = 0, . . . ,∞ ,by

J0 = 1, ∂fJn = iJn−1, (i∂y + ∂ff )Jn = 0 , (5.19)

where f ≡ f(k1) = x− 2k1y (see (5.15)). Note that, by (5.19), the first fewJn’s are given by

J0 = 1, J1 = i(f + γs), J2 =1

2(J21 − δs − 2iy) ,

J3 =1

3!(J31 − 3J1(δs + 2yi) + εs) , (5.20)

with γs, δs, εs arbitrary constants (here the subscript s stands for a simplepole). Note that when we consider the KPI equation they become functionsof time determined by the associate eq. (1.3)). We also write

Hn(x, y) ≡ Jn(x, y, γs = 0, δs = 0, εs = 0, · · ·).Thus

H0 = 1, H1 = if, H2 = −1

2(2iy + f2), H3 =

1

3!(6yf − if3) .


One has the following:

5.2.2. Result IVLet ρ be the solution of the adjoint equation at k1, and n be the greatestinteger for which the integrals∫

Hj−1ρ(x, y)dxdy, j = 0, . . . , n,

exist.Suppose also that DimKerGk1 = 1. In what follows we define theindex Q = Q(k1) ≡ Q1. Then:

i) Q = n.

ii) The Laurent coefficients are related by

J0ρn−1 + · · ·+ Jn−2ρ1 + Jn−1ρ0 + JnΦ = 0. (5.21)

iii) As r2 ≡ x2 + y2 →∞ we have

Φ ≈ i ∂∂f

logHn = i∂

∂flog fn = i

n

f,

andQ = winding number Hn. (5.22)

Hence Q is a topological invariant for the Schrodinger operator thatcounts the number of zeros that Hn(f, y) has as a function of f .

iv) The functions Hn and Jn of (5.19) (5.32) are given by

Hn(f, y) =

√(2iy)n

n!hn

(√i

2yx

), (5.23)

where the hn are the standard Hermite polynomials. Jn are linearcombinations of the former.

The results of [3] correspond to the case n = 1. This is the case of thewell known lumps associated with KPI.

5.2.3. Result V. Double polesAssume that the solution to (4.2) has the following structure around k = k1:

μ(k) = ν(k) +Φ

k − k1+

Ψ

(k − k1)2, (5.24)

where ν(k) is regular at k1 and tends to 1 as k →∞. Note that, as comparedto the general formula (5.4), we simply write Ψ ≡ Ψ2,1. Suppose also thatΨ �= 0 and that DimKerGk1 = 1. Then


i) If u = O(1/r2) then Q2 = 0, where

Q2 =sign(Im k1)

2πi

∫ ∫uΨdxdy.

Let Q ≥ 2.

ii) In the cases below one has the following relationships between μ andits Laurent coefficients:

Φ + J1Ψ = 0 , (5.25)

and also if (with the coefficients ρr from (5.3)):

1. Q = 2⇒ρ0 + J2,dΨ = 0 , (5.26)

where

J1 ≡ J1,d ≡ i(f + γd), J2,d ≡ −1

2(J21 + 2iy + δd) ,

and γ, δd are arbitrary constants (here d stands for double pole).We find it convenient to use a similar notation to that of the simplepole case.

2. If Q = 3 then

ρ1 + (J1 + 2σ)ρ0 +1

3J3,dΨ = 0 , (5.27)

where

J3,d ≡ −(J1 + σ)3 + 3σ2(J1 + σ)− 6iσy + iβ , (5.28)

and β, δd, σ are arbitrary constants.

iii) If Φ,Ψ are related with μ by the above equations then as r2 ≡ x2+y2 →∞ we have

Φ = i∂

∂f1log hn,d +O(1/r

2),

and

Q =1

2πi

∫Γ∞

∂fJn,dJn,d

df = winding number of Jn,d . (5.29)


5.3. SCHRODINGER POTENTIALS

In this section we construct several families of decaying potentials forthe nonstationary Schrodinger operator. Given an assumed pole structure(5.9) of μ around both k1 and k1, the value of all the relevant charges isdetermined. The results of the last section show that under the provisoDimKerGk1 = 1 a system of linear equations, which serve to fix the coeffi-cients of the poles and thus μ(k), is obtained. We begin by considering thesimplest possibility.

5.3.1. 1+1 polesAssume that μ is meromorphic with simple poles at points k1 ≡ a+ ib, k1:

μ(k) = 1 +Φ1

k − k1+

Φ1

k − k1. (5.30)

Then the charges defined by eq. (5.10) are given by

i) Q1 = Q1 = 1. (We use the notation Q1 = Q(k1), Q1 = Q(k1)).

ii) The Laurent coefficients satisfy the system of linear algebraic equations(5.21) with n = 1 : ρ0α = −i(fα + γα)Φα, α = 1, 1 .

iii) Let γ2 = γ1, z = x′ − 2ay′, x′ = x− x0, y′ = y − y0,

x0 =γI a− γR b

b, y0 =

γI2b, (5.31)

where a, b, γR, γI are arbitrary real constants. Then the function (5.30)solves (4.18) with the Schrodinger potential u(x, y) = 2∂xx logF , with

F = z2 + 4b2y′2 +1

4b2. (5.32)

The well known one lump solution is obtained from

u(x, y, t) = 2∂2

∂x2logF.

We say that the solution has charge (index) one.

5.3.2. Simple + double polesAssume that μ has the following structure:

μ = 1 +Φ1

k − k1+

Ψ2

(k − k1)2+

Φ1

k − k1. (5.33)

Suppose also that DimKerGk1 = DimKerGk1 = 1. Then


i) Q1 = 2 .

ii) The Laurent coefficients satisfy the following system of linear algebraicequations

a) At k = k1 (see eqs. (5.25) and (5.26))

Φ1 + J1Ψ2 = 0, ρ0 + J2,dΨ2 = 0 . (5.34)

b) At k = k1 (see eq. (5.21))

ρ1 + J1ρ0 + J2Φ1 = 0 . (5.35)

Note that here we simply write Ψ2,1 ≡ Ψ2.

iii) The function (5.33) solves (4.2) with the Schrodinger potential u(x, y) =2∂xx logF , where

F =(z2 − 4b2y′2 + δR)2 + (2y′(1 + 2bz) +

γIb− δI

)2+1

b2

((z − 1

2b)2 + 4b2y′2 +

1

4b2

), (5.36)

δR, δI being arbitrary real constants.

5.3.3. Two double polesAssume that μ has the following structure:

μ = 1 +Φ1

k − k1+

Ψ2

(k − k1)2+

Φ1

k − k1+

Ψ2

(k − k1)2, (5.37)

where DimKerGk1 = 1. Then

i) Q1 = 3 .

ii) At k = kα, α = 1, 1, the Laurent coefficients satisfy the following systemof linear algebraic equations (obtained from (5.27),(5.28)):

Φα = −J1,αΨ2,α; (5.38)

ρ1,α + (J1,α + 2σα)να +13J3.α,dΨ2,α = 0 , (5.39)

where the notation is defined as follows:

2.1 ≡ 2, 2.1 ≡ 2, 3.1 ≡ 3, 3.1 ≡ 3 .


iii) Ifk1 = k1, β1 = β1, γ1 = γ1 + 2iσ1, σ1 = σ1

the function (5.37) solves (4.2) with the Schrodinger potential deter-mined by

F =(z3 − 12b2y′2z + 3(σ2R − σ2I )z + 12bσIσRy′ − 6σRy′ + βR

)2+(6bz2y′ − 8b3y′3 − 6σIσRz + 6b(σ2R − σ2I )y′ + 6σIy′ − βI

)2+

9

4b2

{(z − σI −

1

2b

)2+ (2by′ − σR)2 +

1

4b2

}×{(z + σI +

1

2b

)2+ (2by′ + σR)

2 +1

4b2

}, (5.40)

where the coordinates z, x′, y′ are the same as those in eq. (5.31) except

x0 =a

b(γI − σR)− (γR + σI), y0 =

γI − σR2b

,

and β is a minor redefinition of the original free parameter β. Thissolution depends on 8 parameters.

5.4. KPI SOLUTIONS

The above class of potentials is related to localized rationally decayingsolutions of the KPI equation when the appropriate temporal dependenceis inserted. Requiring μ(x, y, t, k) to solve also (1.3) implies a particulartime evolution for the constants γ, β, . . . entering in the potential. Here wepresent the solutions corresponding to potentials derived earlier and thenwe study their dynamics.

Solutions to KPI are obtained when the constants of (5.20), (5.28)“evolve” as kα(t) = 0 and

γα(t) = 12k2α, δα(t) = −24ikα, εα(t) = 24i (5.41)

σ(t) = 0, βα(t) = 12(1 + 6kασ), α = 1, . . . , 2N .

5.4.1. Simple polesThe simplest case corresponds to (5.32). Using (5.41) the KPI solutionu(x, y, t) is obtained by setting

x→ x− 12(a2 + b2)t, y → y − 12at (5.42)

in (5.32).


Corresponding to n = 2N simple poles the resulting solution KPIsolution, called the N lump solution, can be obtained. It describes a config-uration that asymptotically is made up of N -humps all of them movinguniformly with distinct velocities vjx = 12(a2j + b

2j ), vjy = 12aj (here

kj ≡ aj + ibj). Upon interaction there is neither a phase shift nor adeflection, hence the asymptotic dynamics of these objects is trivial [11].

5.4.2. Simple + double polesThe KPI solution follows from (5.36) and (5.41) by taking

x→ x− 12(a2 + b2)t, y → y − 12at, δR → δR − 24bt . (5.43)

Study of this KP solution shows that for long times it has two maximaor humps, each of which moves with distinct asymptotic velocities and di-verge from one another, proportional to |t|1/2 (this “anomalous scattering”was first discussed in [13] with particular choice of some of the relevantparameters, and the solution itself was first considered in [12]).

As t→ −∞ the two maxima (+ denotes fast, − denotes slower hump)are located at ⎧⎪⎪⎨⎪⎪⎩

x± ∼ 12(a2 + b2)t±√−24a2tb

+ x0 −1

2b

y± ∼ 12at±√−6tb + y0 ,

(5.44)

and at t→∞ the two maxima are located at{x = 12(a2 + b2)t±

√24bt+ 0(t−1/2)

y = 12at+ y0 .(5.45)

To describe the dynamics of this motion we consider a frame with theorigin at (x0, y0) which moves with velocities vx = 12(a2 + b2), vy = 12a.With respect to this Galilean frame as t → −∞ one of the humps islocated in the first quadrant while the other is the mirror image in the thirdquadrant. As t→∞ both of them travel on the x axis, the first one movingleft, the second right. The asymptotic amplitude is u(x±, y±) = 16b2 forboth maxima as |t| → ∞. The angle Ω the humps get deflected is given by:

Ω = arctan1

2|a| . (5.46)

By properly choosing the free parameter a we can obtain a solution thatscatters with any desired angle. This is in contrast to the situation forstandard lumps where this angle is zero (i.e. there is no phase shift).


5.4.3. Two double polesSolutions to KPI are obtained by taking

x→ x− 12(a2 + b2)t, y → y − 12at, β → β + 12(1 + 6ibσ)t (5.47)

and σ = σR + iσI constant in (5.40). This solution depends on 8 realparameters.

The expression for the solution simplifies if σ = 0, in which case

F = (z3 − 12b2y′2z + 12t+ βR)2 + (8b3y′3 − 6bz2y′ + βI)2

+9

4b2

((z − 1

2b)2 + 4b2y′2 +

1

4b2

)((z +

1

2b)2 + 4b2y′2 +

1

4b2

). (5.48)

This corresponds to a three humped structure with the position of thehumps given asymptotically as |t| → ∞ by

xj ∼ 12(a2 + b2)t+ x0

+ (12t)1/3(cos

(2πj3

)− absin

(2πj3

))+ o(1) (5.49)

yj ∼ 12at+ y0 −(12t)1/3

2bsin

(2πj3

)+ o(1), j = 0, 1, . . . . (5.50)

Thus the humps are located on the conic (x′ − 2ay′)2 + 4b2y′2 = (12t)2/3

moving with uniform speed vx = 12(a2 + b2), vy = 12a and whose semiaxes

are first decreasing and then increasing at a rate |t|1/3. Moreover in themoving frame the humps translate along three straight lines. After collisionthe formulae (5.49)–(5.50) show that the particles follow their original tra-jectories without any scattering or phase shift. Thus there is no scatteringalthough there is nontrivial interaction.

Another subcase of interest is that given by σ = i/(6b), which corre-sponds to

F =(z3 − 12b2y′2z − z

12b2+ βR

)2+(8b3y′3 − 6bz2y′ − 5 y

′

6b+ βI

)2+

9

4b2

{(z +

2

3b

)2+4b2y′2 +

1

4b2

}{(z − 2

3b

)2+ 4b2y′2 +

1

4b2

}. (5.51)

We note that the temporal dependence drops out: (5.51) corresponds to athree-humped KP solution that is stationary (cf. also refs. [13, 14]) whichmoves with uniform velocity vx = 12(a2+ b2) > 0, vy = 12a, and hence canbe regarded as a solitary wave. Thus this solution, unlike the others con-sidered which depend on all the variables z, y′, t, also solves the Boussinesqequation. Solutions that behave asymptotically as a nonlinear superpositionof these multipeaked solitary waves can be obtained in a straightforwardway (see ref. [17]).


6. Rigorous results for the KP equation with decaying initialdata

The analysis of the previous sections assumed the existence and uniquenessof solutions to the relevant integral eqs. (2.5) and (4.2).In this section wecomment on some rigorous results to the direct problem associated withthe KP equation with decaying initial data.

6.1. RIGOROUS RESULTS FOR THE KPII EQUATION WITH DECAYINGINITIAL DATA

We consider eq. (2.5), and recall some standard notation in analysis. Denoteby L∞(IR2), Lp(IR

2, dxdy) ≡ Lp(dxdy), or Lp(IR2, ρdxdy) ≡ Lp(ρdxdy) theset of all measurable functions f(x, y) on the plane with the norms belowfinite; the relevant spaces and norms are defined as

L∞(IR2) = {f ∈ IR2 : ‖f‖∞ ≡ supx,y∈IR2

|f(x, y)| <∞}

Lp(dxdy) = {f ∈ IR2 : ‖f‖p ≡(∫IR2|f(x, y)|pdxdy

) 1p

<∞};

Lp ≡ Lp(ρdxdy) = {f ∈ IR2 :

‖f‖p ≡(∫IR2|f(x, y)|pρ(x, y)dxdy

) 1p

<∞} .

Here p ∈ IR+ and ρ(x, y) is any measurable function on the plane.

6.1.1. Result Ii) The Green’s function can be split as

G(x, y, k) = G∞(x, y, k) +G2(x, y, k), (6.1)

where

G∞(x, y, k) ∈ L∞(IR2), G2(x, y, k) ∈ L2(dxdy) . (6.2)

ii) Assume that u(x, y, 0) ∈ L2(dxdy) ∩ L1(dxdy) and that the condition

M(k) ≡ ‖G∞‖∞ ‖u‖1 + ‖G2‖2 ‖u‖2 < 1 (6.3)

is satisfied. Then there exists a unique solution μ to (2.5) on L∞(IR2)and

‖μ(x, y)‖∞ ≤ 1/(1 −M) . (6.4)


iii) Assume that u(x, y, 0) ∈ L∞(IR2) ∩ L1(dxdy). Define

μ ≡ μ(x, y)√|u(x, y)|,

K(x, x′, y, y′, k) ≡ G(x− x′, y − y′, k)√|u(x, y)|

√|u(x′, y′)|, (6.5)

and consider the transformed equation.

μ(x, y, k) =√|u(x, y)| +

∫dx′dy′K(x, x′, y, y′, k)μ(x′, y′, k). (6.6)

iii.1) the kernel K(x, x′, y, y′, k) is compact, with L2 norm boundedas

‖K‖22 ≤∫dxdy dx′dy′ |K(x, x′, y, y′, k)|2 ≤ M2 (6.7)

M2 ≡ 2(‖G∞‖2∞‖u‖21 + ‖G2‖22‖u‖1‖u‖∞

). (6.8)

iii.2) If the condition M < 1 is satisfied, there exists a unique solutionμ on L2(IR

2) to (6.6), and hence there also exists a unique solutionμ(x, y) to (2.5) in the weighted space

L2

(IR2,

√u(x, y)dxdy

)≡ L2

and

‖μ(x, y)‖2 = ‖μ(x, y)‖2 ≤ ‖u‖121 /(1− M) . (6.9)

iii.3) If M <∞ and no homogeneous modes exist in this space, thereexists a unique solution μ(x, y) to (2.5) in the space L2, regardlessof the size of M .

Proof. We sketch the main ideas. Recall (2.7)

G(x, y, k) =1

(2π)2

∫ ∞

−∞dp

∫ ∞

−∞dq G(p, q) exp[i(px+ iqy)] ,

where

G(p, q) ≡ 1

p2 + 2pk + iq≡ 1

p2 + 2pk + iq(χD + (1− χD))

≡ G∞(p, q) + G2(p, q) .

Here D is a convenient region enclosing the points p = q = 0 and p = −2kR,q = 4kRkI , D

c ≡ R −D, and by χD we denote the characteristic function


of the set D (which is defined under eq. (2.9)). Note that the integrandhas singularities at these points, but the singularities are integrable. Notealso that 1/|p2 + 2kp + iq|2 = O(|p|4 + |q|2)−1 is integrable away from the

singularities, hence G∞(p, q) ∈ L1(dpdq), G2(p, q) ∈ L2(dpdq), and

G∞ ≡1

(2π)2

∫D

dpdqei(px+iqy)

p2 + 2pk + iq; ‖G∞‖∞ ≤

1

(2π)2‖G∞‖1

G2 ≡1

(2π)2

∫Dc

dpdqei(px+iqy)

p2 + 2pk + iq; ‖G2‖2 =

1

2π‖G2‖2

satisfy (6.2).Next, if Gμ stands for the second term on the RHS of (2.5) one has

‖Gμ‖∞ ≡ supx,y∈IR2

∣∣∣∣ ∫ G(x− x′, y − y′, k)uμ(x′, y′, k)dx′dy′∣∣∣∣

≤ ‖μ‖∞ supx,y∈IR2

∫dx′dy′

∣∣∣u(x′, y′)(G∞ +G2)(x− x′, y − y′)∣∣∣

≤ ‖μ‖∞[‖G∞‖∞

∫dx′dy′|u(x′, y′)|

+

(∫|G2|2(x, y)dxdy

∫|u2(x′, y′)|dx′dy′

) 12]≡ ‖μ(x, y)‖∞M,

and the operator G is a contraction on L∞if M < 1.To prove (6.7) notice that if r ≡ (x, y), dr ≡ dxdy, etc., we have∫drdr′|K(r, r′)|2 =

∫drdr′|(G∞ +G2)|2(r − r′)|u(r)||u(r′)|

≤ 2∫drdr′|u(r)||u(r′)|[|G∞(r − r′)|2 + |G2(r − r′)|2]

≤ 2‖G∞‖2∞dr′∫drdr′|u(r)||u(r′)|+ 2‖u‖∞

∫dr|G2(r)|2

∫dr′|u(r′)|

≡ M2 .

Assuming that u(x, y) ∈ L∞(IR2)∩L1(dxdy), it follows that the bound (6.7)applies and the kernel K(x, x′, y, y′) is compact on L2(dxdy); if, besides,M < 1, the RHS of (6.6) is a contraction on L2(dxdy). The rest followsfrom standard properties of compact or contraction operators.

Remark.

1. The basic idea of the proof, namely, the decomposition (6.1), appearsin [4]. With this, the rest follows easily. The results of iii) were alsopointed out in [4], and solvability on L∞(IR2) with the bound (6.4) in[23].


2. As the proof shows, the functions M,M of (6.3) — and hence thebounds (6.4), (6.9) — depend on k. The following result gets aroundthis difficulty:

6.1.2. Result IIi) M(k) is bounded:

M(k) ≤(54‖u‖1‖u‖22C2

) 13 , C ≡ 1

(2π)2

∫ 2π

0

dϕ√| sinϕ| . (6.10)

If the condition54‖u‖1‖u‖22C2 < 1 (6.11)

is satisfied, there exists a unique solution μ to (2.5) on L∞(IR2), and

‖μ(x, y)‖∞ ≤[1− (54‖u‖1‖u‖22C2)

13

]−1.

ii) M (k) is bounded:

M (k) ≤ 3 12 2

56

(‖u‖∞‖u‖21C2

) 13. (6.12)

If the condition

312 2

56

(‖u‖∞‖u‖21C2

) 13< 1 (6.13)

is satisfied, there exists a unique solution μ(x, y) to (2.5) in the weightedspace L2(IR

2,√u(x, y)dxdy) ≡ L2(IR

2), and

‖μ(x, y)‖2 ≤ ‖u‖121

[1− 3 1

2 216 (‖u‖∞‖u‖21C2)

13

]−1.

iii) If ‖u‖∞, ‖u‖1 <∞, and no homogeneous modes exist in the weightedspace L2(IR

2), there exists a unique solution μ(x, y) to (2.5) in thespace L2.

Estimates (6.10), (6.11) are obtained by choosing the region D in asuitable way. We skip the proof. See also [36] regarding the non existenceof exponentially decaying homogeneous modes, and [37] for another boundon M(k).

The solvability of the inverse problem was considered in [23]. The basicresult is the following:

6.1.3. Result IIIAssume that (6.2) is satisfied. Then:


i) There exists g(k) such that

‖g‖∞ <∞ and limkR→∞ g(k) = 0 ;

|F (k)| ≤ ‖u‖11−g(k)(‖u‖1+‖u‖2) . (6.14)

ii) If at the initial time u has two derivatives in

L1(dxdy) ∩ L2(dxdy) ∩ L∞,

thenF (k) ∈ L2(kIdkRdkI) ∩ L∞ ,

and the solution to (2.5) also leads to a potential (2.20) which solvesthe KPII eq. (2.1).

iii) More generally, if at time 0 u has n derivatives in

L1(dxdy) ∩ L2(dxdy) ∩ L∞,

∂(γ1,γ2)u ∈ L1(dxdy) ∩ L2(dxdy) ∩ L∞ (6.15)

for some multi-index (γ1, γ2), then

F (k) = O

(1

1 + |kRkI |γ2 + |kI |γ1), (6.16)

and the solution to (2.5) also leads to a potential (2.20) which solvesthe KPII eq. (2.1).

The reader may wish to consult [23] for the details of the proof.

6.2. RIGOROUS RESULTS FOR THE KPI EQUATION WITH DECAYINGINITIAL DATA

We now consider the KPI equation (4.2).Following [33], we introduce a a function ν(p, y) that satisfies the allied

integral equationν(p, y) = H(p, y) + G(ν) , (6.17)

where H(p, y) and the integral operator G(ν) are defined as

H(p, y) ≡∫dy′u(p, y′)Ξ(p, y − y′) exp

[i(y − y′)(p2 + 2kp)

]G(ν) ≡ ∫

dy′Ξ(p, y − y′) exp[i(p2 + 2kp)(y − y′))

] ∫dlu(l, y)ν(p− l, y)

Ξ(p, y) ≡ (θ(y)θ(∓p)− θ(−y)θ(±p)) . (6.18)


Note that the above equation is formally obtained from (4.2) letting ν bethe Fourier transform of ν ≡ μ− 1 and u(p, y) that of u(x, y) :

u(p, y) ≡∫dxe−ipxu(x, y); ν(p, y) ≡

∫dxe−ipxν(x, y) . (6.19)

To study (6.17) we define the norm

‖ν‖1,∞ ≡ supy

∫dp| ν(p, y)| . (6.20)

6.2.1. Result IVAssume that u(p, y) ∈ L1(dpdy) has small norm, concretely

M ≡∫ ∫

dydp |u(p, y)| < 1 . (6.21)

Then a solution ν ∈ L1(dpdy) to eq. (6.17) exists, is unique and satisfiesthe bound

‖ν‖1,∞ ≤M

1−M . (6.22)

We refer to [33] for the proof. This result has been sharpened, with estimatesfor the derivatives given, in [34].

Note. The results of [33] imply that the Fourier transform of ν(p, y),

ν(x, y) ≡ 1

2π

∫dpeipxν(p, y) ,

exists and is bounded:

‖ν(x, y)‖∞ ≤M

2π(1 −M) .(6.23)

However it does not necessarily follow that ν(x, y) defined by (6.2.1) solveseq. (4.2). The following result fills this gap and establishes existence tosolutions to both (6.17) and (4.2):

6.2.2. Result VConsider the eq. (6.17), with u(p, y) given. Let

u(x, y) ≡ 12π

∫dpeipxu(p, y)

I1(x) ≡∫dy|u(x, y)|, I∞(x) ≡ supy |u(x, y)| .

(6.24)

Assume that:


i) u(x, y) is in L1(dx) for every y and in L1(dy) for every x.

ii) I1I∞ ∈ L 12(dx), i.e., ∫

dx√I1(x)I∞(x) <∞ . (6.25)

iii) Eq. (6.21) holds.

Let ν be the unique solution to (6.17). Then its Fourier transform ν(x, y)solves (4.2), i.e., there exists a solution to (4.2) in the subspace of L∞(IR2)consisting of those functions having a Fourier transform that belongs toL1(dxdy). Moreover, ν(x, y) satisfies the estimate (6.23).

The latter result establishes existence to solutions to both (6.17) and(4.2). This suggests that one can get around the need to consider the alliedintegral eq. (6.17), and work only in physical space. Indeed, one has thefollowing:

6.2.3. Result VIConsider eq. (4.2) with u(x, y) given and satisfying ii) above with (6.25) suf-ficiently small:

∫dx

√I1(x)I∞(x)� 1. Then there exists a unique solution

to (4.2) with bounded norm ‖ν(x, y)‖∞.

Note. Conditions i), ii) above hold if u(x, y) ∈ L∞(IR2)∩L1(dxdy) andI∞(x) ∈ L1(dx). Hence existence and uniqueness to (4.2) are guaranteed if∫

dxdy|u(x, y)| � 1,

∫dx sup

y|u(x, y)| � 1 . (6.26)

The proof will be skipped.

The rigorous theory of the inverse problem was considered in [34]. Thebasic result reads as follows:

6.2.4. Result VIIAssume that the conditions of result IV hold, and that u(p, y) has enoughderivatives in L1(dp dy). Then the equations of the direct and inverse prob-lems (6.17) and (4.20) have both a unique solution, and the potential (4.21)with time dependence (4.22) inserted is real.

The following results improves on the solvability of the inverse problem(for both KPI and KPII) [37]:


6.2.5. Result VIIIAssume that the potential u(x, y) is real with the relevant norm bounded(but not necessarily small). Then, the inverse problem (for both KPI andKPII) is always solvable.

7. On constraints for KP and the associated initial value problem

In this section we study and comment on the difficulties inherent to theKP equations (both for KPI and KPII) due to its nonlocal nature. Tounderstand the onset of these difficulties, we write (1.1) in evolution form,i.e.,

ut + 6uux + uxxx + 3ε2∂−1x uyy = 0 , (7.1)

whereupon it follows naturally that if u and its derivatives decay at infinitythen the solution should satisfy the condition

χ1(y, t) = ∂yy

∫ ∞

−∞u(x, y, t)dx = 0. (7.2)

Indeed, (7.2) has so far been assumed for the initial data u(x, y, t = 0).Note that (7.2) is not preserved in time, and consequently further con-

straints (obtained by differentiation of (7.2) with time) are necessary, e.g.

χ2(y, t) = ∂yy

∫ ∞

−∞dx

(∫ x

−∞dx′ −

∫ ∞

xdx′

)u(x′, y, t) = 0. (7.3)

This procedure yields a hierarchy of constraints χn(y, t) = 0, n = 1, . . . ,∞,which many researchers impose at t = 0. For a more complete discussionon this matter and the relation with conservation laws, we refer the readerto [38]–[40].

We note that in the physical reduction of KP χ1(y, 0) is generically notzero, i.e, χ1(y, 0) �= 0 corresponds to the physical situation. It is importantto decide to what extent are the constraints χn(y, t) = 0, n = 1, . . . ,∞,necessary, and whether there exists a solution corresponding to initial datau(x, y, t = 0) not satisfying (7.2). This delicate problem was addressed in[41]. Note that the difficulty is already present at the linear level, corre-sponding to dropping the term uux of (7.1). As the linear theory is alreadysubstantial enough and encompasses the main difficulties of the problem,we shall concentrate on it henceforth. In this context the nonlinear problemis reducible to the linear one as for such boundary conditions, the linearproblem has all the critical features. Note also that similar issues arise inthe context of one-dimensional problems, e.g. the Sine-Gordon equations inlight-cone coordinates (uxt = sinu). Such i.v.p.’s with initial time disconti-nuity are physically important and generic. It was found in [41] that, even if


χ1(y, 0) �= 0, nevertheless χ1(y, t) = 0 for any t > 0, and the correspondingKP evolution exists. A related issue which will also be discussed is that ofdetermining the natural choice for ∂−1x . It was proven in [41] (see (7.14))that for t > 0 all choices are equivalent, but they produce different resultsat t = 0. In [42] it was shown how the latter discontinuity is resolved in aphysical context (e.g. from water waves) and for more general KP systems.

7.1. THE LINEAR THEORY

A solution to the linear version to (7.1) may be derived formally by usingFourier transforms. We find

u(x, y, t) = −∫ ∞

−∞dp

∫ ∞

−∞dq ei[px+qy+(p3−3ε2q2/p)t]φ(p, q, 0), (7.4)

where −∫∞−∞ denotes the Cauchy principal value integral, and the choice

∂−1x =∫ x−∞ dx

′ −∫∞x dx′ is made to eliminate a delta function δ(p) in the

exponent of (7.4).In order to guarantee that (7.4) is an actual solution of (7.1), the

following conditions are imposed:

(1 + |p3|+ |q2|)φ(p, q, 0) ∈ L1(dpdq),

φ(0, q, 0) ∈ L1(dq), (7.5)

u(x, y, 0) ∈ L1(dxdy) .

This requirement guarantees that u( · , · , t) is of class C3,2 for t > 0, and therelevant derivatives can be carried out under the integral sign and vanish asr →∞. The formal time derivative of this expression brings down a factorof p−1:

∂u

∂t= −∫ ∞

−∞dp

∫dq i

(p3 − 3ε2q2

p

)ei[px+qy+(p3− 3ε2q2

p)t]φ(p, q, 0) , (7.6)

thereby producing a a singularity rendering the existence of ∂u/∂t unclear.Existence is guaranteed requiring the condition

φ(0, q, 0) = 0 . (7.7)

Note. Eq. (7.7) implies both (7.2) and

φ(p = 0, q, t) = 0 , where φ(p, q, t) = φ(p, q, 0)ei(p3−3ε2q2/p)t . (7.8)

Further, u is C1 with respect to t and (7.6) holds with ∂u∂t → 0 as r →∞.


If the condition (7.7) is not satisfied, the issue of whether (7.4) solves(7.1) is much more delicate. Essentially, it amounts to proving that theintegral (7.6) exists, that all derivatives ∂yy, ∂xxx, ∂

−1x and ∂t can be taken

within the integral sign and that the integrals involved in (7.6) commute.The difficulty can be put down to the fact that the integral is neitherabsolutely convergent, nor well defined, in principle, as a Cauchy integral;hence the standard analysis theorems do not apply. In spite of this difficultyin [41] it was proven that for all t > 0 one has the following (note that it issufficient to consider positive times, since we are considering an evolutionequation):

7.1.1. Result IAssume that the conditions (7.5) hold, and that in the vicinity of the originφ(0, q, 0) �= 0 and

|φ(p, q, 0) − φ(0, q, 0)| ≤ g(q)|p|δ for some δ > 0, g ∈ L1 . (7.9)

Then one has:

a) The iterated integrals of (7.6) exist and are equal, i.e.,(−∫ ∞

−∞dp

∫ ∞

−∞dq −

∫ ∞

−∞dq −

∫ ∞

−∞dp

)εi[px+qy+(p3−3ε2q2/p)t]φ(p, q, 0)

p

= 0 . (7.10)

The integral appearing in the right hand side of (7.6) exists for all t asa principal value integral.

b) Formula (7.6) holds, i.e., differentiation with respect to t under theintegral sign in (7.4) is permitted.

c) For all time t > 0 the following formula holds:

−∂−1x uyy(x, y, t) = −∫ ∞

−∞dp

∫dq q2

ei[px+qy+(p3− 3ε2q2

p)t]

ipφ(p, q, 0) .

(7.11)That is, interchange of the x′-integral with those over p and q ispermitted. Here

∂−1x = a

∫ x

−∞−b

∫ ∞

x, a+ b = 1.

d) At time t = 0 one has that

−∂−1x uyy(x, y, 0)

= −∫ ∞

−∞dp

∫dqei(px+qy)

ipφ(p, q, 0) + π(2a− 1)D(y), (7.12)


where

D(y) ≡∫dqeiqyq2φ(0, q, 0) = − 1

2πχ1(y, 0) . (7.13)

Notes.

1) It follows that only in the symmetric case a = b = 12 does (7.11) at the

initial time equal (7.12). On these grounds, it was suggested in [41]that a = b = 1

2 is the natural choice for ∂−1x .

2) The proof of the above statements is given in [41]. Some technicalconsiderations are proven in the appendix at the end of this chapter.Important implications that follow from the proof are the following:

7.1.2. Result IIAssume that the initial data satisfy the conditions (7.5), (7.9), but not(7.2), i.e., φ(0, q, 0) �= 0. Then:

a) At any time t > 0 (7.2) is attained, although nonuniformly, and more-over

limx→∞

∫ x

−xu(x′, y, t)dx′ = 0 pointwise (∀t > 0), (7.14)

b) For large |x|, u(x, y, t) decays weakly as |t/x3| 14 and hence, even withstrongly decaying initial data, the solution does not belong to L1(IR2)for any t > 0. If, besides, u(x, y, 0) belongs to L2(IR2), then u(x, y, t)belongs too to L2(IR2), for all t ≥ 0, and the solution is unique in anL2 setting.

The proof is based in establishing (see the appendix) that the latterintegral behaves as

limx→∞

∫ x

−xu(x′, y, t)dx′ = lim

x→∞

∫dqeiqyJ0(2

√3q2xt)φ(0, q, 0)

=

∫dq eiqyφ(0, q, 0) lim

x→∞ J0(2√3q2xt) = 0 , (7.15)

since J0(2√3q2xt) ∝ (xt)−1/4 as x→∞ and

|J0(2√3q2xt)φ(0, q, 0)| ≤ |φ(0, q, 0)|


(J0 is the standard Bessel function). This also implies by differentiationthat for large x

u(x, y, t)− u(−x, y, t) =∫dq φ(0, q, 0) eiqy

√q2t

xJ1(2

√3q2xt)

=

∣∣∣∣ tx3∣∣∣∣ 14 ∫ dq φ(0, q, 0)eiqy |q| 12 + o

(∣∣∣ tx3

∣∣∣ 14) as x→∞ . (7.16)

Square integrability follows from (7.4) and Parseval’s relationship.

7.1.3. Result IIIAssume that the initial data f(x, y) ≡ u(x, y, 0) satisfy

(1 + x)f ∈ L1(dxdy) ,

and that they are of class C5,4 with the derivatives

fyy, fxxx, fyyyy, fxxxxx ∈ L1(dxdy).

Then (7.9), (7.5) hold, and hence so too results I and II.This is all one needs to establish that the linear KP evolution with

initial data not satisfying (7.2) exists, along with the properties of such anevolution for t > 0.

Several other researchers have revisited this problem. Properties of thesolution in a weak or distributional sense with quickly decaying initial data(concretely data belonging to the Schwartz space) were considered in [43].The issue of whether a time reversal evolution (t < 0) is possible along withits properties was also considered in [43]. It was pointed out that (for KPI)the time derivative of the solution suffers a jump at t = 0:

limt→0+

∂u

∂t(x, y, t) − lim

t→0−

∂u

∂t(x, y, t) �= 0 . (7.17)

Likewise ∂−1x is discontinuous as t→ 0+ and

limt→0+

∫ x

−∞uyy(x, y, t) =

∫ x

−∞uyy(x, y, 0) ,

or, in view of (7.14) ,

limt→0+

∂−1x uyy(x, y, t) =∫ x

−∞uyy(x, y, 0) . (7.18)

We finish this part by making a few remarks in this regard. The function∂u/∂t is only defined (via (7.6)) for t > 0. If this formula is used to extend


its definition to t ≤ 0 then, assuming that the conditions of Result I hold,one can establish the following:

7.1.4. Result IVDefine ∂u∂t (x, y, 0) as the right-hand side of (7.6) at t = 0. Then

i)

∂u

∂t(x, y, 0+)− ∂u

∂t(x, y, 0−)

= 2

(∂u

∂t(x, y, 0+)− ∂u

∂t(x, y, 0)

)= −6πD(y) . (7.19)

ii)

limt→0+

∂−1x uyy(x, y, t)− ∂−1x uyy(x, y, 0) = [2a− 1) + ε2]πD(y). (7.20)

iii) Higher-order time derivatives ∂nu∂tn exist for all n ≥ 2 and t > 0, but

become singular at the initial time.

Proof. Use (7.12) and Lemma 3 of the appendix.

Consequence. ∂u∂t (x, y, t) is discontinuous at t = 0 no matter what choiceof ∂−1x is taken. The choice

a =1− ε22

=

{1, for KPI0, for KPII

renders ∂−1x uyy(x, y, t) continuous for t ≥ 0 but u(x, y, t) with ∂u∂t (x, y, 0) as

defined above does not solve KP at t = 0.If the symmetric choice a = 1

2 is taken, then ∂−1x uyy is discontinuous at

the initial time but∂u

∂t+ 3ε2∂−1x uyy

is continuous and hence u(x, y, t) solves KP for t ≥ 0.

Remark. From a physical perspective one does not expect that differentchoices ∂−1x u arise. Besides, as it has been pointed out, initial time discon-tinuities can be expected from physical requirements [42]. All this suggestsagain that the symmetric choice a = 1

2 is preferred.


7.2. THE NONLINEAR KPII THEORY

Following [41], in this section we show how the above results generalize todeal with the full nonlinear eq. (7.1); for convenience, we take ε2 = 1, i.e.,KPII. We shall assume that

(1 + |p|3 + |q|2)F (k)μ(x, y, k) ∈ L1,

which implies that uxxx and uyy tend to zero as r →∞, and that differenti-ation can be performed under the integral sign in (2.20). With small normthe results of chapter 2 corresponding to the direct side still hold. Insightinto the behavior of (2.20) is obtained when we transform to coordinatesp ≡ −2kR, q ≡ 4kRkI defined in (2.10):

u(x, y, t) =1

4π

∂

∂x

∫ ∫dp dq ei(px+qy)

f(p, q)

pμ

(x, y,−1

2

(p+ i

q

p

))=

1

4π

∫ ∫dp dq ei(px+qy)f(p, q)ζ , (7.21)

where ψ = μx/p, ζ ≡ iμ + ψ and f(p, q, t) ≡ F (−p/2,−q/2p, t). For alltime there exists an eigenfunction μ satisfying μ → 1 as |k| → ∞ whosetemporal evolution corresponds still to formula (2.22), or

f(p, q, t) = f(p, q, 0)ei(p3−3q2/p)t . (7.22)

We note that p→ 0, q fixed, corresponds to kR → 0, kI →∞, whereupon(2.17) yields

μx ∼μ1xikI

= −2piqμ1x(x, y, t) , (7.23)

and consequently ψ is regular as p → 0. The analogue of (7.7) — that wedo not assume in this section — is F (kR, kI) = o(kR). Consequently thepotential u(x, y, t) takes the form

u(x, y, t) =

∫ ∫dp dq ei(px+qy+(p3−3q2/p)t)φ(x, y, p, q, t) , (7.24)

where

φ(x, y, p, q, t) =1

4πf(p, q, 0)ζ

(x, y, t,−1

2

(p+ i

q

p

)). (7.25)

The analogy with the linear problem is clear. Moreover we can formallyevaluate the integral of u(x, y, t). Using (2.17) and (7.15) and the Appendixwe get

limx→∞

∫ x

−xu(x′, y, t)dx′


= − limx→∞

1

4π−∫ ∞

−∞dp

∫dq f(p, q, t)eiqy

(eipx − e−ipx

p

)

=1

4πlimx→∞

∫dq q2eiqyJ0(2

√3q2xt)f(0, q, 0) = 0 . (7.26)

We have just seen, at least in a formal way, that, as happens for thelinear problem, KPII evolution without the condition F (kR, kI) = o(kR),i.e., when the constraints (7.2) are not required, exists. Besides, for t > 0 wehave

∫∞−∞ u(x, y, t)dx = 0. We expect that all that is required for a rigorous

proof is that the small norm condition (6.10), or (6.12), of the previoussection is satisfied.

Next we shall briefly comment on the Hamiltonian of the problem:

H = −∫ ∞

−∞dp

∫dq

{p2|f |2(p, q, t)− 3q2

p2|f |2(p, q, t)

}, (7.27)

which is divergent as it stands. It is regularized as follows;

H = −∫ ∞

−∞dp

∫dq

{p2|f |2(p, q, t)− 3q2

p2

(|f |2(p, q, t)− |f |2(0, q, t)

)}(7.28)

where H is conserved and reduces to the “standard” one if f(0, q, 0) = 0.In order to check that it is the Hamiltonian of the problem, it suffices tocheck that the equations of motion follow from ut =

∂∂xδHδu , which it does.

Other regularizations can be used to give the equations of motion. Sincethe regularizing term is a Casimir, i.e., it commutes with any integral ofmotion, the regularization is not unique.

7.3. NONLINEAR KPI THEORY

The rigorous aspects of KPI evolution without initial value constraintsχ1(y, 0) = 0 have been studied in [44], assuming that the initial data are inthe Schwartz space. The basic result is the following:

7.3.1. Result VAssume that the initial data are in the Schwartz space and that∫

(1 + p2)|u(p, y)|dy � 1 , (7.29)

where

u(p, y) ≡ 1

2π

∫dxeipxu(x, y, 0) . (7.30)

Then a solution u(x, y, t) to KPI (7.1) exists that satisfies


i)

u(x, y, t) ∈ C∞{(x, y, t); t > 0} .

ii) u(x, y, t) satisfies (7.26).

iii) u(x, y, t), with ∂tu(x, y, 0) properly chosen, solves KP for t ≥ 0.

We refer to [44] for the proof.

8. Appendix

In the sequel we assume that

g(p, q),∂g

∂p∈ L1(dpdq) and g(0, q) ∈ L1(dq) . (8.1)

Then one has:Lemma 1

lim|x|→∞

−∫ ∞

−∞dp

∫dq eipx

g(p, q)

ip= π

∫dq g(0, q) .

Proof. Use the decomposition

g(p, q)

p=

g(0, q)

p(1 + p2)+ g ,

where

g(p, q) ≡ 1

p(g(p, q) − g(0, q)

1 + p2) ∈ L1(dpdq) .

Lemma 2

Let

N(x, t) ≡ −∫ ∞

−∞dp

∫dqe

i

(px− ε2q2t

p

)g(p, q)

ip.

Then, as |x| → ∞,

N(x, t) = −2π∫dqJ0(2

√|ε2q2x|t)g(0, q) sign(−ε2x) + o(1) −→

|x|→∞0.

where J0 is the standard Bessel function.

Proof.


With ε2 = 1, we can write the integral as N(x, t) = (1) + (2) + (3),where

(1) ≡ −∫|p|≥1

dp

∫dq ei(px−

q2tp) g(p, q)

ip

(2) ≡ −∫ 1

−1dp

∫dq ei(px−

q2tp) g(p, q)

ip

tend to zero as x→∞ by the Riemann–Lebesgue lemma. Here g ≡ g(p, q)−g(0, q). Finally we consider

(3) ≡ −∫ 1

−1dp

∫dqg(0, q)

ei(px−q2tp)

ip.

We close the latter contour considering an indentation above p = 0 and asemicircular arc with radius one in the upper/lower (for x >, x < 0) halfplane. The integrals over these arcs go to zero exponentially. Hence

(3) = −2π∫dq g(0, q)Res

(ei(px−

q2tp)

p

)p=0

θ(−x) +O(e−|x|)

= −2π∫dq g(0, q)J0(2

√q2|x|t

)θ(−x) +O(e−|x| ) .

By Lebesgue’s theorem (since J0(2√3q2|x|t) ∝ |xt|−1/4 as |x| → ∞) the

claim is obtained.

Lemma 3As t→ 0+ we have

limt→0+

N(x, t) = N(x, 0) − π sign(ε2)∫dq g(0, q) .

Proof. One has

N(x, t) = −∫ ∞

−∞dp

∫dqe−i ε2q2t

p

ip

(eipxg(p, q)− g(0, q)

1 + p2

)+H(t) .

Here

H(t) ≡ −∫ ∞

−∞dp

∫dq e−i

ε2q2tp

g(0, q)

ip(1 + p2)

= −∫ ∞

−∞dp

∫dq e−iε

2q2tp pg(0, q)

i(1 + p2)

= −π sign(ε2)∫dq g(0, q)e−|ε

2|t .


An application of dominated convergence shows that

limt→0+

H(t) = −π sign(ε2)∫dq g(0, q) ,

limt→0+

−∫ ∞

−∞dp

∫dqe−i

ε2q2tp

ip

(eipxg(p, q) − g(0, q)

i(1 + p2)

)= N(x, 0) = −

∫ ∞

−∞dp

∫dqeipxg(p, q)

ip.

On the proof of (7.11)–(7.14)Here we finally sketch the proof of eqs. (7.11)–(7.14). Set

∂−1x uyy(x′, y, t) dx′ = lim

L→∞

(a

∫ x

−L−b

∫ L

x

)uyy(x

′, y, t) dx′ .

Using (7.4) and the fact that φ(p, q, t) ∈ L1(IR2), and taking into accountthat the domain of integration is finite, we obtain(

a

∫ x

−L−b

∫ L

x

)dx′ uyy(x′, y, t)

−−∫ ∞

−∞dp dq

(a

∫ x

−L−b

∫ L

x

)dx′ ei(px

′+qy+p3t− ε2q2tp)φ(p, q, t)q2

= −−∫ ∞

−∞dp

∫dq e

i(px+qy+p3t− ε2q2tp) φ(p, q, t)

ipq2

+−∫ ∞

−∞dp

∫dq

(b eipL + ae−ipL

) φ(p, q, t)ip

q2eiqy+p3t−i ε2q2t

p .

Thus the proof reduces to showing that the last integral goes to zero asL→∞. There are two cases, t = 0 and t > 0. With t = 0 Lemma 1 yields(7.12), while if t > 0 Lemma (2) yields eqs. (7.11) and (7.14).

Acknowledgments

This work was partially supported by the NSF under grant number DMS-0070792 and and by Junta de Castilla-Leon JADZ, SA43/00B in Spain.

References

1. M.J. Ablowitz, H. Segur, J. Fluid Mech., 92, (1979), 6791.2. B.B. Kadomtsev, V.I. Petviashvili Sov. Phys. Doklady, 15, (1970) 539.3. A.S. Fokas and M.J. Ablowitz, Stud. Appl. Math., 69, (1983) 211.


4. M.J. Ablowitz, D. Bar Yaacov, A.S. Fokas, Stud. Appl. Math., 69, (1983) 135.5. S.V. Manakov, Physica D, 3, (1981) 420.6. M.J. Ablowitz and A.S. Fokas, Lecture Notes in Physics 189, Springer Verlag

(Berlin), (1983). See also the contribution of A.S. Fokas and M.J. Ablowitz in thesame Lecture Notes.

7. M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform , SIAM,Phila., PA (1981).

8. M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations andInverse Scattering, Cambridge University Press, Cambridge, UK, (1991).

9. V. Dryuma, Sov. Phys. JETP, 19, (1974) 381.10. J. Satsuma, J. Phys. Soc. Japan, 40 (1976) 276.11. S.V. Manakov, V.E. Zakharov, L.A. Bordag, A.R. Its and V.B. Matveev, Phys. Lett.

A, 63 (1977) 205.12. R.S. Johnson and S. Thompson, Phys. Lett. A, 66 (1978) 279.13. K.A. Gorshov, D.E. Pelinovskii and Yu. A. Stepahyants, JETP, 77 (1993) 237.14. V.M. Galkin, D.E. Pelinovsky, and Yu A. Stepanyants, Physica D, 80 (1995) 246.15. R.S. Ward, Phys. Lett. A, 208 (1995) 203.16. M.J. Ablowitz and J. Villarroel, Physical Review Letters, 78 (1997) 570.17. J. Villarroel and M.J. Ablowitz, Commun. Math. Phys., 207 (1999) 1.18. M.J. Ablowitz and J. Villarroel, Scattering in Pure and Applied Science, Chapter

6.3, R. Pike and P. Sabatier eds., Academic Press (2002) 1792.19. M. Manas and P. Santini, Phys. Lett. A, 227 (1997) 325.20. M.J. Ablowitz, S. Chakravarty, A.D. Trubatch and J. Villarroel, Phys.Lett.A, 267

(2000) 132.21. J. Villarroel and M.J. Ablowitz, SIAM J. Math Anal. (2003, in press).22. Q.P. Liu and M. Manas, J. Nonlinear Sci., 9(1999) 213.23. M. Wickerhauser,Commun. Math. Phys., 108 (1987) 67.24. J. Villarroel and M.J. Ablowitz, Stud. Appl. Math., 109 (2002) 151.25. M. Boiti, F. Pempinelli, A. Progrebkov, and M. Polivanov, Inverse problems, 8

(1992) 331.26. M. Boiti, F. Pempinelli, A. Progrebkov, Inverse problems, 13 (1997) L7.27. M. Boiti, F. Pempinelli, A. Progrebkov, and B. Prinari, Inverse problems, 4 (2001)

937.28. M: Boiti, F. Pempinelli, A. Progrebkov, and B. Prinari, Phys. Lett. A, 285 (2001)

30729. A. Fokas, D. Pelinovsky, C. Sulem, Physica D, 152 (2001) 189.30. A. Fokas, A. Progrebkov, Inverse scattering transform for KPI on the background

of one soliton, preprint.31. L. Faddeev, J. Math.Phys., 4 (1963) 72.32. P. Deift and E. Trubowitz. Comm. Pure and Appl. Math., 32 (1979) 121.33. H. Segur, Mathematical methods in Hydrodynamics and integrability in dynamical

systems (M. Tabor and Y.M. Treve, eds.), AIP conference proceedings 88 (1982)211.

34. X. Zhou, Commun. Math. Phys., 128 (1990) 551.35. A. Degasperis, Nonlinear wave eqs. solvable by the spectral transform, Lectures at

the International School of Physics E. Fermi, (1988), Varenna, Italy.36. P. Grinevich, Lett. Math. Phys., 40 (1997) 59.37. A.S. Fokas, L.Y. Sung, Inverse problems, 8 (1992) 673.38. V. E. Zakharov, E. Schulman Physica D, 1 (1980) 192.39. Z. Jiang, R. K. Bullough, S. V. Manakov, Phys. D, 18 (1986) 305.


40. J. Lin, H. Chen, Phys. Lett. A, 89 (1982) 163.41. M.J. Ablowitz and J. Villarroel, Stud. Appl. Math., 85 (1991) 195.42. M.J. Ablowitz and X-P. Wang, Stud. Appl. Math., 98 (1997) 121.43. M. Boiti, F. Pempinelli and A. Pogrebkov, Inverse Problems 10 (1994) 505.44. A.S. Fokas, L.-Y. Sung, Math. Proc. Camb. Philos. Soc., 125 (1999) 113.45. E. Olmedilla, Physica D, 25 (1987) 330.

PARTIALLY SUPERINTEGRABLE (INDEED ISOCHRONOUS)

SYSTEMS ARE NOT RARE

F. CALOGERO ([email protected])Dipartimento di Fisica, Universita di Roma “La Sapienza”,00185 Roma, Italy, and Istituto Nazionale di Fisica Nucleare,Sezione di Roma

Abstract. We call partially superintegrable (indeed isochronous) those dynamical systemsall solutions of which are completely periodic with a fixed period (“isochronous”) in a partof their phase space, and we review a recently introduced trick that allows to manufacturemany such systems. Several examples are discussed.

1. Introduction

Recently a trick — amounting essentially to a change of independent, andoften as well of dependent, variables — has been introduced [1], that allowsto modify a dynamical system so that the new (modified) system therebyobtained features a lot of completely periodic (indeed isochronous) solutions.This approach has been applied in a number of contexts [2]–[16]. In thispaper we review these results, we report some additional ones, but especiallywe introduce the notion of partially superintegrable (indeed isochronous)dynamical systems and we emphasize the fact, already advertised in thetitle of this paper, that these systems are not rare – indeed, given a genericdynamical system, one can generally associate to it (via the trick) one ormore somewhat analogous systems that are partially superintegrable (indeedisochronous).

We shall now explain this fact, but firstly we must introduce the notionof partially superintegrable (indeed isochronous) system, as employed in thispaper.

Integrable, and superintegrable, systems have a standard definition inthe context of finite-dimensional Hamiltonian dynamics. Let N be thenumber of degrees of freedom of a Hamiltonian system. Then this system isintegrable if there exist N−1 nontrivial functionally independent univalentfunctions of the N canonical variables and of the N conjugate momenta

49


50 F. CALOGERO

that Poisson-commute (namely, their Poisson brackets vanish) among them-selves and with the Hamiltonian, so that, including the Hamiltonian itself,the system features N constants of motion in involution. The Hamiltoniansystem is said to be superintegrable(or perhaps maximally superintegrable– but we will refrain from using such hyperinflated terminology) if thereexist N−1 additional functionally independent univalent functions of theNcanonical variables and of the N conjugate momenta that Poisson-commutewith the Hamiltonian, so that, including the Hamiltonian itself, the systemfeatures 2N − 1 nontrivial constants of motion. It is then easily seen that,for systems that only feature confined motions — and in this paper werestrict attention to such systems — the property of superintegrabilityentails that all motions are completely periodic. Because of this fact, inthis paper any system (including non Hamiltonian systems, and systemswith an infinite number of degrees of freedom) is termed superintegrable ifits generic motions (namely, all, or almost all, its motions) are completelyperiodic; indeed the type of complete periodicity we encounter is moreovercharacterized by a fixed period, independent of the initial data (althoughsome motions may also, or only, be completely periodic with a period whichis an integer submultiple, or an integer multiple, of that basic period),hence these motions should perhaps more properly be characterized asisochronous.

We call a system partially superintegrable (indeed isochronous) if it fea-tures this property, but only in a domain of phase space (rather than inthe entire phase space), a domain which must however be completely open(namely, open in all dimensions). In particular the partially superintegrable(indeed isochronous) systems considered in this paper are characterized bythe property to possess an open domain in phase space such that all themotions evolving from a set of initial data in it are completely periodic withthe same period (isochronous). The measure of such an open domain doesnot vanish; it might, or it might not, be infinite when the measure of theentire phase space is itself infinite (for instance, if the entire phase space isthe two-dimensional Euclidian plane, such a domain might be the exterior,or the interior, of a circle of finite radius).

For instance a well-known superintegrable system is the one-dimensionalmany-body problem characterized by the Hamiltonian [2]

H(p, q) =1

2

N∑n=1

(p2n + ω

2q2n

)+1

4g2

N∑n,m=1,m �=n

(qn − qm)−2 , (1.1)

and correspondingly by the Newtonian equations of motion

PARTIALLY SUPERINTEGRABLE SYSTEMS ARE NOT RARE 51

qn + ω2qn = g

2N∑

m=1,m �=n(qn − qm)−3 . (1.2)

Here and always below ω is a positive constant, ω > 0, and the rest of thenotation is, we trust, self-evident. Indeed, in the real domain, all the solu-tions of these equations of motion are completely periodic (“isochronous”)with period

T = 2π/ω, (1.3)

q(t+ T ) = q(t). (1.4)

This is not quite true in the complex domain, namely if we consider theNewtonian equations of motion

zn + ω2zn = g

2N∑

m=1,m �=n(zn − zm)−3 , (1.5)

which are identical to (1.2) except that we now consider the dependentvariables zn ≡ zn(t) to be complex rather than real (and we also allowthe “coupling constant” g to be complex, while we always consider theconstant ω to be real, indeed, without loss of generality, positive, ω > 0;and of course we always assume the independent “time” variable, t, to bereal). Then all motions, which take of course place in the complex plane,are again completely periodic, but the period may be an integer multiple ofT , see (1.3): indeed also in this case the particle configuration does repeatitself with period T , but the individual particles might exchange their rolesthrough the motion, so that the period of the motion of each individualparticle might end up being an integer multiple of T (this cannot happenin the real case, when the motion takes place on the real line and theordering of the particles cannot change throughout the motion due to thesingular character of the repulsive two-body interaction, see (1.2)) [2].

Hence the many-body problem characterized by the Hamiltonian (1.1)is superintegrable, both in the real and in the complex domains. The moregeneral many-body problem characterized by the following Hamiltonian,

H(p, z) =1

2

N∑n=1

(p2n + ω

2z2n

)+1

4

N∑n,m=1,m �=n

g2nm (zn − zm)−2 , (1.6)

is instead only partially superintegrable, and only if it is considered in thecomplex domain. Indeed it has recently been shown [6] that the Newtonian

52 F. CALOGERO

equations of motion

zn + ω2zn =

N∑m=1,m �=n

g2nm (zn − zm)−3 , (1.7)

that clearly obtain from the Hamiltonian (1.6) (provided of course gnm =gmn, as we hereafter assume), yield a completely periodic motion providedthe initial data fall in an appropriate (open) domain, which however gener-ally does not include only real data. (Incidentally: a remarkable generaliza-tion of this result is reported and proved below, see Sections 2 and 4 andthe Appendix).

These examples present rather special dynamical systems. Let us nowjustify the statement made above (and in the title of this paper), by showingthat to any given dynamical system (with the only qualification that itsequations of motion be analytic, see below) one can generally associateone or more somewhat analogous ones that are partially superintegrable(indeed isochronous) — implying that such partially superintegrable (indeedisochronous) systems are not rare.

Let us use the following notation to denote a generic (but analytic)dynamical system:

ζτ (x, τ) = F [ζ(x, τ);x, τ ] . (1.8)

Here ζ ≡ ζ(x, τ) is the dependent variable, which might be a scalar, avector, a matrix, a tensor, you name it, and that depends of course onthe independent variable τ and might moreover depend on a number ofadditional variables or parameters, here denoted collectively with the vec-tor notation x; the notation ζτ (x, τ) denotes the (partial) derivative ofζ(x, τ) with respect to the independent variable τ ; and the quantity in theright-hand side, F [ζ(x, τ);x, τ ], has of course the same character (scalar,vector,...) as ζ(x, τ), it might depend explicitly on the independent variableτ as well as on the additional variables and parameters x, and it is generallya (nonlinear) functional of the dependent variable ζ(x, τ), say a (nonlinear)function of ζ(x, τ) and possibly of its (partial) derivatives with respect tosome of the parameters subsumed in the notation x (or perhaps also of someintegrals of ζ(x, τ) over these parameters). We are therefore considering atthis stage a fairly general evolutionary system, although in the followingour consideration will be generally restricted to (systems of) Ordinary Dif-ferential Equations (ODEs) and to Partial Differential Equations (PDEs),namely to dynamical systems (especially to those interpretable as many-body problems, such as the examples discussed above) and to evolutionequations of standard type.

We shall however generally assume that the evolutionary equation (1.8)is defined in the complex, namely that not only the dependent variable


ζ(x, τ) take its values in the complex, but that as well the independentvariable τ is complex ; and this of course requires that the derivative ζτ (x, τ)with respect to this variable be well defined, namely that the general contextbe that of analytic (of course, not necessarily singularity-free, namely notnecessarily entire) functions of the complex variable τ . In this context it isthen a general consequence of the existence/uniqueness/analyticity theoremfor analytic evolution equations that the solution ζ(x, τ) of the initial-valueproblem (namely, the solution ζ(x, τ) that evolves out of a given “initialdatum” ζ(x, 0)) for the evolutionary system (1.8) is a holomorphic functionof the complex variable τ in a disk of the complex τ -plane centered at τ = 0and having a nonvanishing radius, say ρ > 0 — provided, as we hereafterassume, the right-hand side of (1.8) is nonsingular initially, namely forτ = 0 and ζ = ζ(x, 0). The value of ρ is determined by the properties ofthe right-hand side of (1.8) in the neighborhood of τ = 0 and ζ = ζ(x, 0),and it is generally larger the smaller (in modulus) that right-hand side isin that neighborhood; and one can always identify a sufficiently small valueof this radius ρ, so that there exist an open set of initial data ζ(x, 0) suchthat all the corresponding solutions ζ(x, τ) be holomorphic functions of theindependent (complex) variable τ in the disk of radius ρ centered, in thecomplex τ -plane, at the origin, τ = 0.

Let us now introduce the following change of independent variable:

z(x, t) = ζ(x, τ) (1.9)

withτ ≡ τ(t) = η [exp(iωt)− 1] /(2i), (1.10)

where η and ω are two constants, with the essential requirement that ω bereal (indeed, without loss of generality, positive, ω > 0; while η is at thisstage an arbitrary complex number, but see below).

Now we consider z ≡ z(x, t) to be the new dependent variable, and tthe new independent variable, with the important restriction to considerthis latter real, indeed to interpret it hereafter as the “time” variable. Ofcourse the new dynamical problem reads now, in self-evident notation,

z(x, t) = (ηω/2) exp(iωt)F [z(x, t);x, τ(t)] . (1.11)

Before proceeding further, let us note that the change of variable (1.9)with (1.10) entails that the “initial” data for the evolutionary systems (1.8)and (1.11) are very simply related:

z(x, 0) = ζ(x, 0). (1.12)

And now our conclusion: we saw above that all the solutions ζ(x, τ)of (1.8) evolving from an (appropriately chosen, but certainly) open set of

54 F. CALOGERO

initial data are holomorphic functions of the independent (complex) vari-able τ in a disk of (sufficiently small but certainly) positive radius ρ, ρ > 0,centered, in the complex τ -plane, at the origin, τ = 0. But, via (1.9) with(1.10), this immediately implies that all the corresponding solutions z(x, t)of the new dynamical problem (1.11) are completely periodic functions withperiod T , see (1.3), of the new, real, “time” variable t,

z(x, t+ T ) = z(x, t), (1.13)

provided there holds the simple inequality

|η| < ρ. (1.14)

Indeed the relation (1.10) clearly entails that τ is a periodic function oft with period T , see (1.3), and moreover that as t varies over one period,say from 0 to T , τ travels counterclockwise, from τ = 0 back to τ = 0, onthe circle in the complex τ -plane the diameter of which, of length |η|, hasone end at τ = 0 and the other at τ = iη (hence its center at τ = iη/2);this, via the condition (1.14), guarantees that it remains within the disk ofradius ρ centered at τ = 0, namely that it stays in the region in which thesolution ζ(x, τ) is holomorphic in τ ; and this of course entails, via (1.9),that z(x, t) is completely periodic in t, see (1.13).

Clearly, once this argument is understood, its conclusion becomes essen-tially trivial. It is moreover clear that such a conclusion might also obtainif a different change of independent variable had been chosen instead of(1.10), provided it also implied that τ , as a function of the real variable t,be periodic and confined to a neighborhood of the origin, τ = 0. But whatmakes this argument interesting in spite of its triviality is the possibilitythat the new dynamical system (1.11) obtained in this manner be itselfinteresting. Generally a necessary, if not sufficient, condition for this to bethe case is the requirement that the new evolutionary equation, see (1.11),be autonomous (namely, not feature an explicit time dependence). As wewill see, a virtue of the particular change of dependent variable (1.10) is togreatly facilitate the attainment of this goal, although to achieve this it isoften required to perform an additional change of dependent variable, in theguise of the introduction of a prefactor depending (generally exponentially)on t in the relation among the new and old dependent variable, see (1.9).The prefactor should of course have itself the property to be periodic int with the appropriate period. For instance the change of (dependent andindependent) variable

zn(t) = exp(−iωt)ζn(τ) (1.15)

withτ = [exp(2iωt)− 1] /(2iω) (1.16)


entails that the Newtonian equations of motion (1.7) correspond to thefollowing (quite analogous) set of ODEs:

ζ ′′n =N∑

m=1,m �=ng2nm (ζn − ζm)−3 , (1.17)

where of course the appended primes denote differentiations with respectto the independent variable τ . And this fact was key, utilizing the argu-ment we just described, to proving [6] the partial superintegrability (indeedisochronicity) of the Newtonian equations of motion (1.7), as defined above.

This concludes our presentation of the main idea of this paper. In thefollowing Sections, by reviewing the results it has yielded thus far andby presenting some additional findings, we hope to convince the readerthat this approach, in spite (or perhaps because) of the trivial characterof the observation that underlies its applicability, is remarkably effectiveinasmuch as it leads to the identification of many interesting evolutionarysystems that possess lots of completely periodic (indeed isochronous) solu-tions, and of simple prescriptions to manufacture many more such systems.The potential applicative interest of such an approach is, we trust, self-evident, although of course its actual relevance shall have to wait for futuredevelopments in order to be fully demonstrated. Below, in Section 2 wereview some results on many-body problems, in Section 3 we report onthe existence of “nonlinear harmonic oscillators”, in Section 4 we reportsome results on other isochronous oscillators, in Section 5 we exhibit twelveexamples of single nonlinear ODEs with periodic (indeed isochronous) so-lutions, in Section 6 we tersely show, via a single example, how this sameapproach can be used to “deform” nonlinear evolution PDEs so that theypossess many periodic (indeed isochronous) solutions, and finally in Section7 we tersely offer some final remarks. Except when we present new findingsthe treatment is generally limited to a terse description of results, withreferences to the literature for their derivation as well as for more detaileddiscussions of them. The paper is completed by an Appendix where somecomputations are reported, the presentation of which in the body of thepaper would have excessively interfered with the flow of the discourse there.

2. Many-body problems

In this Section we review instances of partially superintegrable (indeed isoch-ronous) many-body problems. Several other such systems can be found in[2], although the full implications of the trick were understood only as thatbook [2] was being completed and are therefore only partially reflectedthere. Of course many other models can be easily manufactured via thetechnology described above and exemplified below.

56 F. CALOGERO

We need not repeat here the analysis [6] of the many-body problemcharacterized by the Newtonian equations of motion (1.7), except to notethat a detailed study of that model outside of the region of phase space inwhich it behaves as a superintegrable (indeed isochronous) system is alsoquite interesting [10], although it exceeds the scope of this paper. Let usmoreover note that, in Section 4, a nontrivial extension of this system isalso shown to be partially superintegrable (again, provided it is consideredin the complex ). It is the many problem characterized by the Hamiltonian

H(p, z) =1

2

N∑n=1

(p2n + ω

2z2n

)

+1

4

N∑n,m=1,m �=n

[g2nm (zn − zm)−2 +

1

2γnm (zn − zm)−4

](2.1)

correspondingly by the Newtonian equations of motion

zn + ω2zn =

N∑m=1,m �=n

[g2nm (zn − zm)−3 + γnm (zn − zm)−5

]. (2.2)

Indeed the treatment given in the following section entails that the sameresult obtains even for the more general dynamical systems characterizedby equations of motion

zn + ω2zn =

K∑k=0

fk,n(z), (2.3a)

provided one retains the freedom to assign a sufficiently large value of thereal constant ω and the functions fk,n(z) of the N variables zn satisfy thesame scaling property as featured by the right-hand sides of (2.2),

fk,n(λz) = λ−3−2kfk,n(z), (2.3b)

and are otherwise essentially arbitrary (but analytic in the N variables zn).Next we report the (Hamiltonian [2]) N -body problem in the plane

characterized by the Newtonian equations of motion

�rn = ωk ∧�rn + 2N∑

m=1,m �=nr−2nm(αnm + α

′nmk ∧ )

·[�rn(�rm · �rnm) +�rm(�rn · �rnm)− �rnm(�rm ·�rm)

]. (2.4)

Here the N two-vectors �rn ≡ �rn(t) identify the positions of the moving


point-particles in a plane which for notational convenience is immersedin three-dimensional space, so that �rn ≡ (xn, yn, 0); k is the unit three-

vector orthogonal to that plane, k ≡ (0, 0, 1), so that k∧ �rn ≡ (−yn, xn, 0);�rnm ≡ �rn − �rm, hence r2nm ≡ �rnm · �rnm ≡ (xn − xm)2 + (yn − ym)2;superimposed dots denote of course time derivatives; ω is as usual a positiveconstant; and the “coupling constants” αnm, α

′nm are a priori arbitrary

(of course real ; a sufficient condition for this system to be Hamiltonian isthe requirement that these constants be symmetrical in their two indices,αnm = αmn, α

′nm = α′mn [2]).

To treat this many-body problem it is generally convenient to identifythe real “physical” plane in which the points �rn ≡ (xn, yn, 0) move with thecomplex plane in which the complex numbers zn ≡ xn + iyn move. Indeedvia this correspondence these equations of motion take the following simplerform:

zn − iωzn = 2N∑

m=1,m �=nanmznzm/(zn − zm) (2.5)

withanm = αnm + iα

′nm. (2.6)

Then one notes that, via the change of independent variable

zn(t) = ζn(τ), (2.7)

τ = [exp(iωt)− 1] /(iω), (2.8)

which is the simple version of the trick appropriate to this case, and entailsthe following very simple relations among the initial data for zn and ζn,

zn(0) = ζn(0), zn(0) = ζ′n(0), (2.9)

the equations of motion (2.5) become

ζ ′′n = 2N∑

m=1,m �=nanmζ

′nζ′m/(ζn − ζm), (2.10)

where of course primes denote differentiations with respect to the (com-plex) independent variable τ . And thereby one easily concludes [3] thatthe N -body problem (2.5) is partially superintegrable (indeed isochronous),featuring motions that are completely periodic (indeed isochronous) withperiod T = 2π/ω provided they evolve from an appropriate open set ofinitial data, which, as can be easily guessed by looking at the right-handside of (2.10) and by recalling the simple relation (2.9) among the initialdata, are characterized by small (in modulus) values of the initial velocities

58 F. CALOGERO

zn(0) and by large (in modulus) values of the initial interparticle distanceszn(0) − zm(0). Moreover, let us again note that the detailed analysis [11]of this N -body problem outside of this region of phase space is as wellquite interesting, although it exceeds the scope of this paper. And let usalso point out that, in special cases, the system is actually superintegrableindeed solvable: for instance if all the coupling constants are unity, anm = 1(in which case it was suggested [17] that the model deserves to be considereda “goldfish”; although extension of this honorary title to the general system(2.5) was also later suggested [11] ), and in some other cases as well [9] [16].

The next example we report in this section is a deformed version of theclassical N -body gravitational problem [12], characterized by the equationsof motion

�rn + iω�rn + 2ω2�rn =

N∑m=1,m �=n

Mm(�rm − �rn)r−3nm, (2.11)

where the N three vectors �rn denote the positions in three-dimensional(ordinary) space of N point particles of masses Mn (actually, the restric-tion to three-dimensional space is merely for convenience, to make con-tact with physical reality, see below; the results hold independently of thenumber of dimensions of ambient space). Of course here rnm denotes thestandard Euclidian distance among the n-th and m-th particle, r2nm =[(�rn − �rm) · (�rn − �rm)].

For ω = 0 clearly this N -body problem, (2.11), coincides with the clas-sical (of course Hamiltonian) N -body gravitational problem. We considerinstead the case with positive ω, ω > 0, when the equations of motion (2.11)define a deformed version of the classical N -body gravitational problem;note that in this deformed case the motion takes necessarily place in thecomplexified three-dimensional space, due to the second term in the left-hand side of (2.11). It is then convenient to introduce [12] the followingchange of dependent and independent variables (the version of the trickappropriate to this case):

�rn(t) = exp(−2iωt)�ρn(τ), (2.12)

τ = [exp(3iωt) − 1] /(3iω). (2.13)

It is then plain that the equations of motion (2.11) take the new form

�ρ ′′n =N∑

m=1,m �=nMm(�ρm − �ρn)ρ−3nm, (2.14)

that only differs from (2.11) due to the absence of the “ω-terms” in the left-hand side, and of course because the superimposed dots are now replaced


by appended primes, which obviously signify differentiations with respectto the new (complex) independent variable τ , see (2.13). And it is then easyto prove [12], via the approach outlined above, that the system (2.11) ispartially superintegrable (indeed isochronous), namely that there is an openset of (complex) initial data for this problem that yield motions completelyperiodic (indeed isochronous) with period T = 2π/ω. The behavior of thissystem for initial data that are outside this domain is an interesting openproblem.

The last example we report in this section is characterized by the follow-ing equations of motion (see [2, Exercise 5.6.20]; note the change of notationfrom, and the correction of the misprint in, eq. (5.6.20-41a) there):

zn − i(4 + p)ωzn − 2(2 + p)ω2zn = Fn(z), (2.15)

where we assume p to be a positive integer, p = 1, 2, 3, . . ., the constantω to be as usual real (indeed, without loss of generality, positive, ω >0), we indicate with z the N -vector the components of which are the Ndependent variables zn ≡ zn(t), and we assume the N functions Fn(z) tobe holomorphic at zn = 0 and to satisfy the scaling property

Fn(λz) = λ1+pFn(z). (2.16)

These equations of motion (that include of course the Newtonian equa-tions of motion of an M -body problem in S-dimensional space, with N =MS) define a system that is partially superintegrable (indeed isochronous),inasmuch as there is, in the neighborhood of the equilibrium configurationz = z = 0, a ball of initial data, of nonvanishing volume in the phasespace (having 4N real dimensions) of this system, such that all the motionsevolving from it are completely periodic, with period T = π/ω if p is even,with period T = 2π/ω if p is odd. This conclusion is easily proved viathe trick consisting of the following change of dependent and independentvariables:

zn(t) = (exp(2iωt) ζn(τ), (2.17)

τ = [exp(ipωt)− 1] /(ipω), (2.18)

whereby the equations of motion (2.15) with (2.16) become (in self-evidentnotation)

ζ ′′n = Fn(ζ). (2.19)

Note incidentally that the gravitational case treated above, see (2.11), maybe considered a special case of (2.15), by relinquishing the restriction topositive values of the parameter p; indeed (2.11) is clearly a special case of

60 F. CALOGERO

(2.15) with (2.16) and p = −3; and indeed, for this value p = −3, there isas well correspondence among (2.12,2.13) and (2.17,2.18), up to a (merelynotational) change of the sign of ω.

3. Nonlinear harmonic oscillators

In this section we report tersely a result obtained with V. Inozemtsev [14]via the approach outlined above: a rather remarkable finding, as indicatedby the title of this section, that might appear an oxymoron since most physi-cists consider, in association with the substantive “oscillator”, the adjectives“linear” and “harmonic”, and conversely “nonlinear” and “unharmonic”, assynonymous. Let us emphasize that the systems considered in this sectionmust be considered superintegrable, not just partially superintegrable.

The starting point is the fact that the matrix evolution equation

Y ′′(τ) = c [Y (τ)]3 , (3.1)

is integrable, and more importantly, that all its solutions are meromor-phic functions of the independent variable τ , for all (finite) values of thiscomplex variable [18]. Here the dependent variable Y ≡ Y (τ) is a matrixof arbitrary rank, c is an arbitrary scalar constant, and of course primesdenote differentiations with respect to the independent variable τ .

We now use the following version of the trick:

M(t) = exp(iωt)Y (τ), (3.2)

τ = [exp(iωt)− 1] /(iω), (3.3)

where as usual ω is a real (say, without loss of generality, positive, ω > 0)constant. It is then plain that the matrix M(t) satisfies the matrix ODE

M(t)− 3iωM (t)− 2ω2M(t) = c [M(t)]3 , (3.4)

while the fact that all the solutions of (3.1) are meromorphic functionsof the independent variable τ clearly entails, via (3.2) with (3.3), that allthe nonsingular solutions of this matrix ODE are completely periodic withperiod T = 2π/ω,

M(t+ T ) =M(t). (3.5)

But clearly the matrix evolution equation (3.4) represents a collection ofnonlinear (linear plus cubic) oscillators, while the result we just mentionedjustifies calling them harmonic (indeed, for arbitrary initial conditions, theyall oscillate with a single period, except for the exceptional cases in whichthey diverge: nothing could be less cacophonous, namely more harmonic,


featuring a single note, of course with all its harmonics). So, the title ofthis section is justified, in spite of its ossimoronic appearance.

This finding can be given a more remarkable look by using appropriateparametrizations [20] [2] [14] of the matrix M(t), which are preserved bythe evolution equation (3.4) and transform this matrix ODE into a systemof coupled oscillator equations that may take various interesting forms,for instance, as systems of coupled rotation-invariant evolution equationssatisfied by an arbitrary number of vectors in a space with an arbitrarynumber of dimensions, say (in self-evident notation) [14]

�znm − 3iω�znm − 2ω2znm = cN∑n′=1

M∑m′=1

�znm′(�zn′m′ · �zn′m) (3.6)

with n = 1, . . . , N , m = 1, . . . ,m, or

�znm − 3iω�znm − 2ω2znm = cN∑n′=1

M∑m′=1

�zn′m′(�zn′m′ · �znm), (3.7)

with n = 1, . . . , N , m = 1, . . . ,m. These vectors �znm ≡ �znm(t) are ofcourse complex, as entailed by the evolution equations they satisfy, see thesecond term in the left hand side of (3.6) and (3.7); but real equations canof course be easily obtained by introducing the real and imaginary partsof these vectors, say �znm(t) = �xnm(t) + i�ynm(t).

Let us also note that modified versions of these Newtonian equationsof motion can also be exhibited [14] which are not only rotation-invariant,but as well translation-invariant, while retaining the property of completeisochronicity of all their nonsingular solutions.

In these examples of applications of the trick an important underly-ing idea was to restrict attention to autonomous (i.e., time-independent)evolution equations, and more specifically to evolution equations that areautonomous in both avatars, before and after application of the trick,namely both in the version with the real independent variable t (“time”)as well as in the related version with the complex independent (“time-like”) variable τ ; indeed the main focus of some recent papers has been theidentification (and/or investigation) of certain ODEs [7] [8], or of classes ofODEs [5] [13], or of (classes of) PDEs [4] [5], possessing such a property.But this restriction is not necessary for the applicability of this approach.Indeed in the following section we indicate how information can be fairlyeasily evinced, via this approach, on the behavior — in particular, theexistence of completely periodic (“isochronous”) solutions emerging out ofcertain open domains of initial data — of certain interesting, autonomous,systems of ODEs describing a “physical” evolution as it unfolds over thereal time t, even though the versions of these systems of ODEs obtained

62 F. CALOGERO

via the trick and characterizing the evolution as function of the complexvariable τ are themselves no more autonomous.

4. Isochronous oscillators

The main new idea — described at the end of the preceding section — thatunderlies the results reported in this section was developed in collaborationwith Jean-Pierre Francoise [15]. We report here the main findings, referringto the recent joint paper with Francoise for proofs and comments, includinga discussion of how such results could (or rather could not) be obtained bymore standard techniques [15].

Let us consider the system of N nonlinear ODEs

zn − irnωzn = fn(z), (4.1)

where zn ≡ zn(t) are the N dependent variables, t is of course the realindependent variable (“time”), ω is a positive number, ω > 0, that sets thetime scale and to which we associate as usual the period T = 2π/ω, the Nnumbers rn are real and rational,

rn =pnqn

� 0 (4.2)

with pn, qn two coprime integers and qn > 0, and each of the N functionsfn(z) of the N components zn of the N -vector z is arbitrary except for therequirements to be holomorphic in each of the complex variables zn nearz = 0 and to actually vanish at z = 0 faster than linearly,

limδ→0

[δ−1fn(δz1, . . . , δzm, . . . , δzN )

]= 0, (4.3)

and moreover to satisfy there the N conditions

limε→0

[ε−1−rnfn(εr1z1, . . . , εrmzm, . . . , εrN zN )

]= finite. (4.4)

If the N rational numbers rn are either all nonnegative or all nonpositive,no additional conditions are required; if instead both positive and negativenumbers rn are featured by the system (4.1), then one must additionallyrequire that all the N functions fn(z) such that the corresponding ratio-nal numbers rm have one of the two signs (namely, are either positive ornegative) depend polynomially on all the coordinates zm.

Provided the functions fn(z) are so restricted, one can then prove [15]that there exists a positive number Z > 0 such that the N conditions onthe “initial data” zn(0),

|zn(0)| < Z, (4.5)


entail that all the corresponding solutions zn(t) of (4.1) are completely

periodic with period (at least) T = qT = 2πq/ω, where q is the minimumcommon multiple of the N integers qn. One therefore concludes that, forsmall enough initial data, the system (4.1) with (4.2) and the conditionson the functions fn(z) specified above (see (4.3) and (4.4), as well as thesentence following (4.4)), retains the isochronicity property of its linearpart, namely of the system that obtains from (4.1) by equating its left-handside to zero, the solutions of which read obviously

zn(t) = zn(0) exp(irnωt) (4.6)

and are therefore completely periodic (indeed isochronous) with period T ..Let us emphasize that, as discussed in [15], this result includes the Hamil-tonian case, which is characterized by an even N , N = 2M , by “oppositelypaired” rn’s, say rM+m = −rm,m = 1, . . . ,M, and by functions fn(z)obtainable by differentiation from a single function h(z) as follows:

fm(z) =∂h(z)

∂zM+m,

fM+m(z) = −∂h(z)

∂zm, m = 1, . . . ,M. (4.7)

As a second example of application of the same type of approach —and there are of course many others, as made clear by these examples —let us now consider the Hamiltonian many-body problem characterized bythe Newtonian evolution equations (see (2.2))

zn + ω2zn =

N∑m=1,m �=n

[g2nm (zn − zm)−3 + γnm (zn − zm)−5

]. (4.8)

The version of the trick we now apply reads

zn(t) = exp(−iωt)ζn(τ), (4.9)

τ = [exp(2iωt) − 1] /(2iω). (4.10)

It is easily seen that this change of dependent and independent variablesentails the following relations among the “initial data”:

zn(0) = ζn(0), zn(0) = ζ′n(0)− iωζn(0), (4.11)

as well as the following evolution equations for the new dependent variableζn(τ):

ζ ′′n =N∑

m=1,m�=n

[g2nm (ζn − ζm)−3 + γnm (1 + 2iωτ) (ζn − ζm)−5

], (4.12)

64 F. CALOGERO

where the primes denote of course differentiations with respect to the newindependent variable τ .

Although these evolution equations, (4.12), are not autonomous (unlessall the “coupling constants” γnm vanish), it is nevertheless still fairly ob-vious, and indeed easy to prove (see Appendix) that, provided the initialdata ζn(0), ζ

′n(0) satisfy the following conditions,

|ζn(0)− ζm(0)| > R, (4.13)∣∣ζ ′n(0)∣∣ < V, (4.14)

with R chosen sufficiently large and V chosen sufficiently small (see (8.18)with (8.13), and (8.19), in the Appendix for an explicit version of suchconditions), then the solutions ζn(τ) of (4.12) are holomorphic in τ in thedisk |τ | ≤ 1/ω, hence these quantities, considered via (4.10) as functions

of the real variable t, are all periodic with period T = π/ω, hence thecorresponding (via (4.9) and (4.10)) solutions zn(t) of (4.8) are completely

antiperiodic with the same period T , zn(t+ T ) = −zn(t), hence completelyperiodic (indeed isochronous) with period T = 2π/ω, zn(t + T ) = zn(t).Note that the conditions we just mentioned, (4.13) and (4.14), define anopen set of initial data ζn(0), ζ

′n(0) for the problem (4.12), to which there

corresponds via (4.11) an open set of initial data zn(0), zn(0) for the originalproblem (4.8). However generally this open set of initial data zn(0), zn(0)need not include only real data.

Before concluding this section let us also note that the Newtonianequations of motion (4.8) obviously admit the (exact, explicit) solution

zn(t) = a cos(ωt) + b sin(ωt) + cn (4.15)

with a and b arbitrary constants and the N constants cn required to satisfythe algebraic system

ω2cn =N∑

m=1,m �=n

[g2nm (cn − cm)−3 + γnm (cn − cm)−1

]. (4.16)

The investigation of the behavior of this model, see (4.8), for initial datathat are outside the domain identified above, see (4.13) and (4.14), is aninteresting open problem.

5. Nonlinear ODEs

In this section we exhibit, with minimal commentary, twelve single nonlin-ear ODEs and their solutions, which are generally isochronous (or quasi


isochronous, see below). These results, elementary as they are, are per-haps interesting thanks to their pedagogical value, as examples useful totest numerical integration routines, as tools to investigate the behavior ofmore general nonlinear systems which possess periodic solutions but arenot amenable to exact treatment, and as additional illustrations of theefficacy of a technology (to identify evolution equations possessing lots ofisochronous solutions) which is likely to prove fruitful in various applicativecontexts, in particular in the mathematical modeling of cyclic phenomena,which are a common feature of the world we live in. Here, however, wedo not elaborate on how this technique works, since we feel this has beensufficiently illustrated above; we merely exhibit twelve examples which haveindeed all been arrived at via such a technique (although they could ofcourse also be obtained otherwise).

Notation: t is the independent variable (hereafter assumed to be real:“time”), and superposed dots denote differentiations with respect to thisvariable t. The ODEs are generally written below in their neater version;additional arbitrary constants can of course be introduced via trivial trans-formations, for instance by multiplying by constant factors the independentand dependent variables or by adding a constant to the latter. The evolu-tion equations written below are generally complex, hence the dependentvariables are as well generally complex ; this entails the possibility (left as anoption for the diligent reader) to reformulate these equations as two coupledreal ODEs by introducing the standard representations of a complex num-ber via its real and its imaginary parts, or via its amplitude and its phase.But in some cases the equations are real, hence their solutions may be realas well. Whenever a square (or cubic) root is written, it is understood thatthe results are valid for anyone of its two (or three) determinations; thisfact justifies our usage sometimes below of the plural to denote some ofthe displayed solutions, or equations, since more than one is entailed bythe possibility to choose different determinations of the square (or cubic)roots.

Example 1. The real ODE

2xx− 3x2 = x2(1− x2) (5.1)

has the general solution

x(t) =sinh(a)

cosh(a)− cos(t− t0). (5.2)

Here a and t0 are two arbitrary constants (real if the solution is real). Notethat this solution is periodic with period 2π (and it is nonsingular in thereal case). Also note that this ODE, (5.1), is the Newtonian equation of

66 F. CALOGERO

motion yielded, in the standard manner [2], by the Hamiltonian

H(x, p) =1

2

[c p2x3 +

1 + x2

c x

], (5.3)

where c is an arbitrary constant (c �= 0) that does not appear in the equationof motion (5.1) and p is the “canonical momentum” conjugated to the“canonical variable” x.


x sin(x) sin(2x) = x2 sin(3x) + sin3(x) cos(2x) (5.4)


x(t) = arctan

[sinh(a)

cosh(a)− cos(t− t0)

]. (5.5)

Here a and t0 are two arbitrary constants (real if the solution is real). Notethat, in the real case, this solution is periodic with period 2π (and it isof course nonsingular). Also note that this ODE, (5.4), is the Newtonianequation of motion yielded, in the standard manner [2], by the Hamiltonian

H(x, p) =1

2

[c p2 sin2(x) sin(2x) +

1

c sin(2x)

], (5.6)



x sin(x) sin(2x) = x2 sin(3x) + sin3(x) (5.7)


x(t) = arctan

[sin(a)

cos(a)− cos(t− t0)

]. (5.8)

Here a and t0 are two arbitrary constants (real if the solution is real).Note that, in the real case, this solution is nonsingular, and it is periodicwith period 2π. [Indeed this formula, (5.8), entails that q(t) vanishes att = t0 + nπ, q(t0 + nπ) = 0, and that [q(t0 + nπ)] (−)n+1 > 0, hence themaxima and minima of q(t) are alternatively attained at t = t0 + nπ, n =0, 1, 2, 3, . . . and they have the same values, namely qmin = q(t0 + 2mπ) =

− arctan[

sin(a)1−cos(a)

]< 0, qmax = q(t0 + π + 2mπ) = arctan

[sin

1+cos(a)

]> 0,

m = 0, 1, 2, . . . ; here we take of course always the principal value of the


arctan function, namely the value in the interval between 0 and π, and weassumed for simplicity (without significant loss of generality) that 0 < a <π2 namely 0 < sin(a) < 1].Also note that this ODE, (5.7), is the Newtonian equation of motion

yielded, in the standard manner [2], by the Hamiltonian

H(x, p) =1

2

[c p2 sin2(x) sin(2x) +

cot(2x)

c

], (5.9)


Example 4. The complex ODE

z + z = (z + iz)3 (5.10)

has the general solutions

z(t) = a exp(−it) + [i+ b exp(−2it)]1/2 , (5.11a)

z(t) = a exp [−i(t− t0)] +{i+ b exp [−2i(t− t0)]

}1/2. (5.11b)

In (5.11a) a and b are two arbitrary complex constants; the expression

(5.11b) corresponds to (5.11a) with a = a exp(−it0), b = b exp(−2it0)and displays a feature of the solution clearly entailed by the autonomouscharacter of the ODE (5.10). Note that these solutions, (5.11a), are pe-riodic with period 2π unless they are singular ; a necessary and sufficientcondition to exclude this (for real t) is validity of the inequality |b| �= 1;moreover, if and only if |b| > 1 these solutions are antiperiodic with periodπ, z(t+ π) = −z(t).

This ODE, (5.10), possesses moreover the special solution

z(t) = c exp(−it), (5.12)

with c an arbitrary complex constant. This solution is clearly nonsingularand periodic with period 2π (antiperiodic with period π). It is not a specialcase of the general solution (5.11).


z + 3iz + z = −4(z − iz/2)3 + 3z2/z (5.13)

has the general solutions

z(t) =

{i+ a exp(it) +

[{i+ a exp(it)}2 + b2 exp(2it)

]1/2}1/2

, (5.14)

68 F. CALOGERO

as well as the special solution

z(t) = c exp(it/2). (5.15)

Here a, b and c are arbitrary complex constants. In (5.14) one may of course

replace t with t− t0 provided one also replaces a, b with a = a exp(it0), b =b exp(it0), and likewise in (5.15) one may replace t with t − t0 providedone also replaces c with c = c exp(it0/2). Clearly the special solution (5.15)in nonsingular and periodic with period 4π (antiperiodic with period 2π).The general solutions (5.14) are as well periodic: if b = 0 the solution (5.14)

vanishes or becomes simply z(t) = ±{2 [i+ a exp(it)]}1/2, in which case itis periodic with period 2π if |a| < 1 , periodic with period 4π (antiperiodicwith period 2π) if |a| > 1, singular if |a| = 1. If b �= 0 the general solutions(5.14) are singular for real t if and only if one of the two quantities

F±(a, b) =∣∣∣a2 + b2∣∣∣2 − [

|a|2 + |b|2 ± 2 Im(ab∗)]

(5.16)

vanishes, F+(a, b) = 0 or F−(a, b) = 0 ; otherwise they are periodic, withperiod 2π or 4π depending on the signs of the two quantities F±(a, b), see(5.16).


z − 3iz − 2z = (z − iz) z (5.17)


z(t) = 2a exp(it) tan [−ia exp(it) + b] (5.18)

and the special solutionz(t) = c exp(it). (5.19)

Here a, b and c are three arbitrary complex constants (a �= 0). Clearlythe special solution (5.19) is periodic with period 2π (and entire), and thegeneral solution (5.18) is as well periodic with period 2π unless it is singular(and conditions on the constants a, b sufficient to exclude this eventualityare obvious). It is also clear how to display the property of the generalsolution (5.18), respectively of the special solution (5.19), that correspondsto the autonomous character of the ODE (5.17): replace t with t− t0 anda with a = a exp(it0) respectively c with c = c exp(it0).


z + iz + 2z = (z + iz)3z (5.20)

has three general solutions, which coincide with the three roots of thefollowing cubic equation in z:

z3 + az exp(2it) + b exp(3it) − 2 = 0 (5.21)


where a, b are two arbitrary complex constants. The coefficients of this cubicequation are clearly periodic functions of t with period 2π, hence the set ofits three roots is as well periodic in t with period 2π, while each of the threeroots is periodic with period 2π, 4π or 6π due to the possibility that, throughthe time evolution, different roots exchange, as it were, their roles. It isleft for the diligent reader, using if need be the explicit expressions of thesolutions of cubic equations, to ascertain the limitations on the constantsa, b which are necessary and sufficient to guarantee that the solution of(5.20) belong to one of these three periodicity regimes. These limitationsidentify open domains of the space of values of these complex constants(and correspondingly of the initial data, say z(0), z(0), which would beassigned in the context of the initial-value problem for (5.20)), domains theboundaries of which correspond to values of the complex constants a, b (orcorrespondingly of the initial data z(0), z(0)) that yield solutions of (5.20)which become singular at some real value of t.

This ODE, (5.20), possesses moreover the special solution

z(t) = c exp(it) (5.22)

where c is an arbitrary complex constant. This solution is clearly nonsin-gular and periodic with period 2π (antiperiodic with period π).

Note that in (5.21) one can replace t with t− t0, provided one replacessimultaneously a with a = a exp(2it0) and b with b = b exp(3it0); likewisein (5.22) one can replace t with t− t0 provided one replaces simultaneouslyc with c = c exp(it0).


z − 5iz − 6z = z2 (5.23)


z(t) = 6 exp(2it)℘(a− i exp(it); 0, g3), (5.24a)

z(t) = 6 exp [2i(t− t0)]℘(a− i exp [i(t− t0)] ; 0, g3). (5.24b)

In (5.24a) a and g3 are two arbitrary complex constants, and ℘(u; g2, g3)is the Weierstrass elliptic function; the expression (5.24b) corresponds to(5.24a) with a = a exp(−it0), g3 = g3 exp(6it0). Note that this solution,(5.24a), is periodic in t with period 2π, unless it is singular ; a conditionnecessary and sufficient to exclude that this happen for real t is provided bythe inequality |a− up − 2nω1 − 2mω2| �= 1, where n,m are two arbitraryintegers, ω1, ω2 are the two semiperiods of the Weierstrass elliptic function℘(u; 0, g3), and up = up(g3) is the argument at which the Weierstrass elliptic

function ℘(u; 0, g3) has a (double) pole, [℘(up; 0, g3)]−1 = 0.

70 F. CALOGERO

Example 9. The complex ODEs

z − 5iz − 4z = (z + iz) z

{z2 +

[z4 + 4(z + iz)

]1/2}(5.25)

have the general solution

z(t) = −a exp(it) tan[(a3/3) exp(3it)− b

](5.26)

and the special solutions

z(t) = [c exp(−3it)− 1/4]−1/3 . (5.27)

In (5.26) a, b are two arbitrary complex constants; to display the invari-ance of this solution under time translations one can of course replace twith t − t0 and a with a = a exp(it0). This solution, (5.26), is clearlyperiodic with period 2π, unless it is singular ; this happens for real t onlyif |a|3 = 3 |b+ (2n+ 1)π/2| for some integer value of n. In the specialsolutions (5.27) c is as well an arbitrary complex constant, which should bereplaced by c = c exp(−3it0) if one wished to replace t with t − t0. Thesesolutions are singular if and only if |c| = 1/4; otherwise they are periodic,with period 2π/3 if |c| < 1/4, with period 2π if |c| > 1/4.

Example 10. The complex ODE.

z − 5iz − 4z = (z + iz) z3 − (z + iz)2 z−1 (5.28)


z(t) = a exp(it){b exp

[−a3 exp(3it)

]− 3

}−1/3. (5.29)

In this formula a and b are two arbitrary complex constants; to display theproperty of this solution that corresponds to the translation invariance ofthe ODE (5.28) one can of course replace t with t− t0 and simultaneously awith a = a exp(it0). This solution, (5.29), is periodic with period 2π unlessit is singular, and clearly conditions sufficient to exclude that this happen

are either |b| > 3 exp(|a|3

)or |b| < 3 exp

(− |a|3

).

Example 11. The complex ODEs

z − iz = z2 [1 + α/ log(z)] /z (5.30)

have the general solutions

z(t) = exp{a exp [exp(it) + b]1/(1−α)

}, α �= 1 (5.31a)

z(t) = exp {a exp [b exp(it)]} , α = 1. (5.31b)


Our usage of the plural in characterizing the ODEs (5.30) as well as thesolutions (5.31a) is motivated by the possibility to choose different deter-minations of log(z) in (5.30) and of the power 1/(1−α) in (5.31a). In theseformulas, (5.30) and (5.31a), α is an arbitrary complex constant (of coursethe same!); in (5.31) a, b are two arbitrary complex constants. In (5.31a)one may of course replace t with t − t0 provided one also replaces a witha = a exp [it0/(1 − α)] and b with b = b exp(−it0), and likewise in (5.31b)one may replace t with t−t0 provided one also replaces b with b = b exp(it0).Clearly the solution (5.31b) is nonsingular and periodic with period 2π.The solution (5.31a) is nonsingular, indeed entire, if α = 1− 1/p with p anarbitrary positive integer ; otherwise these solutions, (5.31a), are singular,at a real value of t, if and only if |b| = 1. If |b| > 1 these solutions, (5.31a),are periodic with period 2π (irrespective of the value of the constant α),while if instead |b| < 1 they are periodic only if α is a real rational number,say α = m/n with m,n two coprime integers, with period 2 |m− n|π.

Example 12. The complex third-order ODE...z − 9iz − 26z + 24iz = z(z − 2iz) (5.32)


z(t) = 12 exp(2it)℘(a − i exp(it); g2, g3), (5.33a)

z(t) = 12 exp [2i(t− t0)]℘(a− i exp [i(t− t0)] ; g2, g3), (5.33b)

and the special solutionz(t) = c exp(2it). (5.34)

In (5.33a) a, g2 and g3 are three arbitrary complex constants, and ℘(u ; g2, g3)is the Weierstrass elliptic function; the expression (5.33b) corresponds to(5.33a) with a = a exp(−it0), g2 = g2 exp(4it0), g3 = g3 exp(6it0). Notethat this solution, (5.33a), is periodic with period 2π, unless it is singular ;a condition necessary and sufficient to exclude that this happen for realt is provided by the inequality |a− up − 2nω1 − 2mω2| �= 1, where n,mare two arbitrary integers, ω1, ω2 are the two semiperiods of the Weier-strass elliptic function ℘(u; g2, g3), and up = up(g2, g3) is the argumentat which the Weierstrass elliptic function ℘(u; g2, g3) has a (double) pole,[℘(up; g2, g3)]

−1 = 0. The special solution (5.34) (where c is of course anarbitrary complex constant) is clearly nonsingular and periodic with periodπ (antiperiodic with period π/2).

6. Nonlinear PDEs

In the previous sections various applications of the trick to manufactureevolution equations featuring many periodic (indeed isochronous) solutions

72 F. CALOGERO

have been reported in the context of (single, or systems of) ODEs. The factthat this approach is as well applicable to PDEs was pointed out in [4] andelaborated in [5]. In this section we merely outline the main idea, on thebasis of a single example.

Consider the hierarchy of (C-integrable) nonlinear evolution PDEs

ϕτ = {R [ϕ; ξ]}m (ϕ2ϕξξ), m = 0, 1, 2, . . . , (6.1)

where ϕ ≡ ϕ(ξ, τ) is the dependent variable, ξ and τ are the independentvariables, subscripted variables denote partial differentiations, for instanceϕτ ≡ ∂ϕ(ξ, τ)/∂τ , and the “recursion operator” R [ϕ; ξ] is defined by theformula

R [ϕ; ξ] = ϕ2(∂/∂ξ)2ϕ(∂/∂ξ)−1ϕ−2, (6.2)

so that the first three PDEs of this hierarchy, (6.1), read as follows:

ϕτ = ϕ2ϕξξ, (6.3)

ϕτ =1

2ϕ2(ϕ2)ξξ, (6.4)

ϕτ = ϕξξξξϕ4 + 6ϕξξξϕξϕ+ 4ϕ

2ξξϕ

3 + 7ϕξξϕ2ξϕ

2. (6.5)

The fact that (6.1) is a hierarchy of C-integrable nonlinear evolution PDEslinearizable via an appropriate hodograph transformation is well-known[21].

Now introduce the following change of dependent and independent vari-ables (the “trick”):

w(x, t) = exp(iωt)ϕ(ξ, τ), (6.6a)

ξ = x exp(irωt), (6.6b)

τ =[exp {i [m+ 1 + r(m+ 2)]ωt} − 1

]×{i [m+ 1 + r(m+ 2)]ω

}−1, (6.6c)

where ω is an arbitrary positive constant and r is an arbitrary real andrational number (possibly zero). Note that this change of variables, (6.6),depends on the positive integer m that identifies the particular equationof the hierarchy (6.1) to which it shall be applied, that it reduces to theidentity (up to a notational change) if ω vanishes, and that generally itentails that old and new dependent variables coincide “initially”,

w(x, 0) = ϕ(x, 0). (6.7)


It is now easy to verify that the new dependent variable w(x, t) satisfiesthe evolution PDE

wt − iωw − irωxwx = {R [w;x]}m (w2wxx). (6.8)

This PDE, (6.8), is autonomous (i.e., there is no explicit t-dependence),nor does it feature any explicit x-dependence if r = 0. It is a “deformation”of (6.1), characterized by the two parameters ω and r: when ω vanishes,ω = 0, it reduces to (6.1) (up to trivial notational changes); when ω doesnot vanish (recall we assumed it is positive), it presumably features lots ofperiodic solutions (or rather, solutions that feature interesting periodicityproperties in the real “time” variable t), since the solution w(x, t) of thisPDE, (6.8), can be obtained via ((6.6) from the solution ϕ(ξ, τ) (having thesame initial datum, see (6.7)) of (6.1), and clearly to every solution ϕ(ξ, τ)of (6.1) that is meromorphic in τ there corresponds a solution w(x, t) of(6.8) that is periodic in the (real) “time” variable t (unless it is singular— a nongeneric happening). The periodicity to be expected is of coursecharacterized by the basic period T = 2π/ω, but integer multiples of itmight come into play if r is not an integer, see (6.6c). Similar, but morerich, periodicity properties are moreover entailed for solutions w(x, t) of(6.8) that correspond to solutions ϕ(ξ, τ) of (6.1) featuring rational branchpoints as functions of τ .

7. Outlook

The observation that partially integrable (indeed isochronous) systems — asdefined in this paper— are not rare might be considered surprising by some,and instead trivial by others. Be it as it may, it leads to the identificationof many dynamical systems and evolution equations that feature lots ofcompletely periodic (indeed isochronous) solutions. Such systems are likelyto be of applicative interest in many contexts, and, at least in some cases[10] [11], their investigation turned out to be remarkably rewarding alsofrom the theoretical point of view. Much, clearly, remains to be done totake full advantage of the potentialities of this approach.

A final remark, on the distinction among partially superintegrable andsuperintegrable systems. Obviously a partially superintegrable system canbe promoted to become a superintegrable one by appropriately restrictingthe phase space in which the system evolves — or equivalently, in thecontext of the initial-value problem, by restricting the domain within whichits initial conditions are to be assigned. We prefer to keep the distinctionamong the two types of systems, and always imagine that every system isallowed to live in its natural phase space.

74 F. CALOGERO

8. Appendix

The result we prove in this Appendix is the existence of two positive num-bers R and V , such that all N components of the solution ζ(τ) of the(second-order, nonautonomous) evolution equations (4.12) are holomorphicfunctions of the complex variable τ in the disk |τ | ≤ 1/ω provided the “ini-tial data” ζ(0), ζ ′(0) are restricted by the conditions (4.13) and (4.14). Herethe underlined symbol ζ denotes of course an N -vector, ζ≡ (ζ1, . . . , ζN ).

To prove this result we take advantage of the following standard Theo-rem (see for instance Section 12.21 of [22]): let a system of analytic evolutionequations read

w′j = Fj(w, τ), j = 1, . . . , J, (8.1)

with the solution w ≡ w(τ) (where w ≡ (w1, . . . , wJ) is of course a J-vector)characterized by the simple initial datum

w(0) = 0. (8.2)

The functions Fj(w, τ) are analytic in all their J +1 arguments, and holo-morphic at w = 0 and τ = 0 (namely where the initial data are assigned, see(8.2)). Then the J components wj(τ) of the solution w(τ) are holomorphicfunctions of the complex variable τ (at least) in the disk |τ | ≤ ρ, with thepositive quantity ρ bounded below by the formula

ρ > a [1− exp {−b/ [(J + 1)aF ]}] , (8.3)

where the positive quantities a and b are characterized by the requirementthat all the functions Fj(w, τ) be holomorphic in all their J +1 argumentsprovided

|τ | ≤ a (8.4)

and|wj | ≤ b, j = 1, . . . , J, (8.5)

while the quantity F is defined by the formula

F = maxj,k=1,...,J|τ |≤a; |wk|≤b

|Fj(w, τ)| . (8.6)

Note that the right-hand side of (8.3) is of course an increasing function ofa and of b, and a decreasing function of F , hence the lower bound (8.3) isvalid a fortiori if we underestimate a and b, and we overestimate F , as weshall do below.

To apply this Theorem to our case we set

J = 2N (8.7)


and

wn(τ) = ζn(τ)− ζn(0), wN+n(τ) = α[ζ ′n(τ)− ζ ′n(0)

], n = 1, . . . , N, (8.8)

with α a positive constant, α > 0, the value of which will be convenientlyassigned below. The condition (8.2) is then automatically satisfied, whileour evolution equations, (4.12), can now be identified with the standardevolution equations (8.1) by setting

Fn(w, τ) = ζ ′n(0) + α−1wN+n (8.9a)

FN+n(w, τ) = αN∑

m=1,m �=n

(g2nm [ζn(0) − ζm(0) + wn − wm]−3

+ γnm (1 + 2iωτ) [ζn(0)− ζm(0) + wn − wm]−5), (8.9b)

n = 1, . . . , N.

We then note, first of all, that we can make the assignment a = ∞(since clearly these functions Fj(w, τ) are nonsingular for all values of τ),hence we replace (8.3) with

ρ > b/ [(2N + 1)F ] , (8.10)

where we also used (8.7).Next we set, say,

b = R/4, (8.11)

an assignment that is clearly adequate, via (4.13), to guarantee that all thefunctions Fj(w, τ), see (8.9), be holomorphic in all their arguments wj inthe ball characterized by the restrictions (8.5).

Now we must find an upper bound to F , see (8.6). Clearly via (8.9),(4.14), and (4.13) with (8.5) and (8.11), we get

F < max{V +R/(4α), αRΩ2/ [2(2N + 1)]2

}, (8.12)

with

Ω =4 (2N + 1)

R3

⎧⎪⎨⎪⎩(N − 1) maxn,m=1,...,Nn �=m

[2∣∣∣g2nm∣∣∣+ |γnm|R2

]⎫⎪⎬⎪⎭1/2

. (8.13)

To get the last formula, (8.13), we used the fact that, since we only needto consider values of τ in the disk |τ | ≤ 1/ω, we can use the maximization|1 + iωτ | ≤ 2.

76 F. CALOGERO

The maximum in (8.12) must be taken by choosing one or the otherargument in the square bracket; we now choose α so that these two termscoincide, namely we set

α =2(2N + 1)2 V

RΩ2

⎧⎨⎩1 +[1 +

(RΩ

2(2N + 1)V

)2]1/2⎫⎬⎭ , (8.14)

hence we get

F <V

2

⎧⎨⎩1 +[1 +

(RΩ

2(2N + 1)V

)2]1/2⎫⎬⎭ . (8.15)

Insertion of this expression and (8.11) in (8.10) yields

ρ >R

2(2N + 1)V

⎧⎨⎩1 +[1 +

(RΩ

2(2N + 1)V

)2]1/2⎫⎬⎭ , (8.16)

and it is then plain that the condition we must prove,

ρ ≥ 1/ω, (8.17)

is entailed by the two inequalities

Ω < ω, (8.18)

V <R (ω2 − Ω2)

4 (2N + 1)ω. (8.19)

Clearly the first, (8.18), of these two inequalities can always be satisfied,see (8.13), by choosing a sufficiently large value of R, and then the sec-ond, (8.19), of these two inequalities can as well be satisfied by choosing asufficiently small value of V . Q. E. D.

References

1. F. Calogero, “A class of integrable hamiltonian systems whose solutions are(perhaps) all completely periodic”, J. Math. Phys. 38, 5711-5719 (1997).

2. F. Calogero, Classical many-body problems amenable to exact treatments, LectureNotes in Physics Monograph m 66, Springer, 2001.

3. F. Calogero and J.-P. Francoise, “Periodic solutions of a many-rotator problem inthe plane”, Inverse Problems 17, 1-8 (2001).

4. F. Calogero and J.-P. Francoise, “Periodic motions galore: how to modify nonlinearevolution equations so that they feature a lot of periodic solutions”, J. NonlinearMath. Phys. 9, 99-125 (2002).


5. F. Calogero, “Differential equations featuring many periodic solutions”, in: Geom-etry and integrability (edited by L. Mason and Y. Nutku), London MathematicalSociety Lecture Notes, vol. 295, Cambridge University Press, 2002 (in press).

6. F. Calogero, “Periodic solutions of a system of complex ODEs”, Phys. Lett. A293,146-150 (2002).

7. F. Calogero, “On a modified version of a solvable ODE due to Painleve”, J. Phys.A: Math. Gen. 35, 985-992 (2002).

8. F. Calogero, “On modified versions of some solvable ODEs due to Chazy”, J. Phys.A: Math. Gen. 35, 4249-4256 (2002).

9. F. Calogero, “Solvable three-body problem and Painleve conjectures”, Theor. Math.Phys. 133, 1443-1452 (2002).

10. F. Calogero and M. Sommacal, “Periodic solutions of a system of complex ODEs.II. Higher periods”, J. Nonlinear Math. Phys. 9, 1-33 (2002).

11. F. Calogero, J.-P. Francoise and M. Sommacal, “Periodic solutions of a many-rotatorproblem in the plane. II. Analysis of various motions”, J. Nonlinear Math. Phys.(in press).

12. F. Calogero, “A complex deformation of the classical gravitational many-body prob-lem that features a lot of completely periodic motions”, J. Phys. A: Math. Gen. 35,3619-3627 (2002).

13. F. Calogero and J.-P.- Francoise, “Nonlinear evolution ODEs featuring manyperiodic solutions”, Theor. Math. Phys. (in press).

14. F. Calogero and V. I. Inozemtsev, “Nonlinear harmonic oscillators”, J. Phys. A:Math. Gen. 35, 10365–10375 (2002).

15. F. Calogero and J.-P. Francoise, “Isochronous motions galore: nonlinearly cou-pled oscillators with lots of isochronous solutions”, Proceedings of the Workshopon Superintegrability in Classical and Quantum Systems, Centre de RecherchesMathematiques (CRM), Universite de Montreal, September 16-21, 2002 (in press).

16. F. Calogero, “General solution of a three-body problem in the plane”, J. Phys. A:Math. Gen. 36, 7291–7299 (2003).

17. F. Calogero, “The “neatest” many-body problem amenable to exact treatments (a“goldfish”?)”, Physica D 152-153, 78-84 (2001).

18. V. I. Inozemtsev, “Matrix analogues of elliptic functions “, Funct. Anal. Appl. 23,323-325 (1990) [Russian original : Funct. Anal. Pril. 23, 81-82 (1989)].

19. M. Bruschi and F. Calogero, “On the integrability of certain matrix evolutionequations”, Physics Lett. A273, 167-172 (2000).

20. M. Bruschi and F. Calogero, “Integrable systems of quartic oscillators”, PhysicsLett. A273, 173-182 (2000).

21. N. Euler, M. L. Gandarias, M. Euler and O. Lindblom, “Auto-Hodograph Transfor-mations for a Hierarchy of Nonlinear Evolution Equations”, J. Math. Anal. Appl.257, 21-28 (2001).

22. E. L. Ince, Ordinary Differential Equations, Dover, New York, 1956.

INITIAL-BOUNDARY VALUE PROBLEMS FOR

LINEAR PDES: THE ANALYTICITY APPROACH

A. DEGASPERIS ([email protected])Dipartimento di Fisica, Universita di Roma “La Sapienza”,Roma, ItalyIstituto Nazionale di Fisica Nucleare, Sezione di Roma

S. V. MANAKOV ([email protected])Landau Institute for Theoretical Physics, Moscow, Russia

P. M. SANTINI ([email protected])Dipartimento di Fisica, Universita di Roma “La Sapienza”,Roma, ItalyIstituto Nazionale di Fisica Nucleare, Sezione di Roma

Abstract. It is well-known that one of the main difficulties associated with any methodof solution of initial-boundary value problems for linear PDEs is due to the presenceof boundary values which cannot be arbitrarily assigned. To deal efficiently with thisdifficulty, we have recently proposed two alternative (but interrelated) methods in Fourierspace: the Analyticity approach and the Elimination by Restriction approach. In this workwe present the Analyticity approach and we illustrate its power by studying the well-posedness of initial-boundary value problems for second and third order evolutionaryPDEs, and by constructing their solution. We also show the connection between theAnalyticity approach and the Elimination by Restriction approach in the particular caseof the Dirichlet and Neumann problems for the Schrodinger equation in the n-dimensionalquadrant.

1. Introduction

It is well-known that one of the main difficulties associated with Initial-Boundary Value (IBV) problems for linear PDEs of the type

L(�, ∂∂t)u(x, t) = f(x, t), x ∈ V ⊂ IRn, t > 0, (1.1)

where � = ( ∂∂x1 , . . . ,∂∂xn

), L is a constant coefficients partial differentialoperator, f(x, t) is a given forcing and u(x, t) is the unknown field, which

79


80 A. DEGASPERIS, S.V. MANAKOV, P.M. SANTINI

may be asked to satisfy Dirichlet, or Neumann, or Robin, or mixed, orperiodic boundary conditions on ∂V , is due to the presence of unknownBoundary Values (BVs) in any method of solution. To deal efficiently withthis difficulty, we have recently proposed two alternative (but interrelated)methods in Fourier space: the Analyticity approach and the Elimination byRestriction (EbR) approach.

The first step, common to both methods, consists in rewriting the PDE(1.1), defined in a space-time domain D, in the corresponding Fourier space,using the Green’s formula. The PDE in Fourier space takes the form of alinear relation among the Fourier Transforms (FTs) of the solution, of theinitial condition and of a set of BVs, only a subset of which is given a priori.The important observation here is that this relation is always supplementedby analyticity requirements on all the FTs involved, as a consequence ofthe geometric properties of the space-time domain D.

The second step is where the two methods differ from each other; oncethe problem is formulated in Fourier space, we propose the following twoalternative strategies.

i) The Analyticity approach consists in using systematically the analyt-icity properties of all the FTs involved in the above relation to derivea system of linear equations which allows one to express the unknownBVs in terms of the known ones, and therefore to solve the problem.

ii) The Elimination by Restriction approach consists, instead, in applyingto the above linear relation in Fourier space a suitable annihilationoperator, which eliminates all the unknown BVs; a new transform istherefore generated, which is well-suited to the specific IBV problemunder scrutiny. The inversion of this new transform (if it exists) leadsto the solution.

The Analyticity approach is inspired by Fokas’s recent discovery of theglobal relation, obtained first within the x− t transform approach [1] andmore recently using differential forms [2]. The use of the global relationto study the well-posedness and to solve IBV problems is illustrated, forinstance, in [3], [4], [5]. In [5], in particular, general results on the well-posedness of IBV problems for dispersive 1 + 1 dimensional equations ofarbitrary order are discussed.

Our main contribution to the method consists, after formulating theIBV problem in Fourier space, in imposing systematically the analyticityproperties of all the Fourier transforms involved in the problem, to derive acascade of analyticity constraints which allow one to express the unknownBVs in terms of the known ones, and therefore to solve the problem. Inparticular, Fokas’ global relation appears, in the methodology we propose,as a “zero residue condition” for the FT of the solution.

IBV PROBLEMS FOR LINEAR PDES 81

The Analyticity approach we propose is very elementary and, aboveall, has the great conceptual advantage to originate itself from a singleguiding principle: satisfying the analyticity properties of all the Fouriertransforms involved in the problem. It is the type of approach that can beeasily taught in University courses, combining nicely standard PDE theorytools, like the Green’s formula and the Fourier transform, with elementarynotions in Complex Functions theory.

The essential aspects of the Analyticity approach were first presentedby the authors at the Workshop “Boundary value problems” in Cambridge,December 2001, inside the programme: “Integrable Systems”. The methodis systematically presented for the first time in this paper, with illustrationsof IBV problems of various type (Dirichlet, Neumann, mixed, periodic) forsome second and third order classical PDEs of the Mathematical Physics:the Schrodinger, the heat and the linear Korteweg-de Vries equations. Alsoits connections with the EbR approach are illustrated here, in the particularexample of the Schrodinger equation in the n-dimensional quadrant. Ageneral account of the EbR approach is given in [6]. A different approach,valid for semibounded domains, has been recently presented in [7]. A generalreview of the basic spectral methods of solution of IBV problems for linearand soliton PDEs is presented in [8].

Here below, §2 is devoted to the presentation of the Analyticity ap-proach, while §3 is dedicated to its application to some IBV problems forsecond and third order evolutionary PDEs in 1+1 and in n+1 dimensions.In §4 we finally discuss the connections between the Analyticity approachand the EbR approach.

2. The Analyticity Approach

2.1. THE FOURIER TRANSFORM AND ITS ANALYTICITY PROPERTIES

The natural FT associated with the space-time domain D = V ⊗ (0,∞) (inshort: FTD) is defined by

F (k, q) =

∫Ddxdte−i(k·x+qt)F (x, t) (2.1)

for any smooth function F (x, t), (x, t) ∈ D, assuming that F (x, t) → 0fast enough if t → ∞; where k = (k1, · · · , kn) ∈ IRn, q ∈ IR and k · x =∑nj=1 kjxj . Its inverse:

F (x, t)χD(x, t) =∫IRn+1

dkdq

(2π)n+1ei(k·x+qt)F (k, q) (2.2)

reconstructs F (x, t) in D and zero outside, where χD(x, t) is the charac-teristic function of the domain D: χD(x, t) = 1, (x, t) ∈ D, χD(x, t) =


0, (x, t) /∈ D (therefore: χD(x, t) = χV (x)H(t), where H(t) is the usualHeaviside (step) function).

If the space domain is the whole space: V = IRn, the FTD (2.1) isdefined in A = IRn ⊗ Iq, where Iq is the closure of the lower half q-planeIq, analytic in q ∈ Iq, ∀k ∈ IRn and exhibits a proper asymptotic behaviorfor large q in the analyticity region. If the space domain V is bounded, theFTD acquires strong analyticity properties in all the Fourier variables: it isdefined in A = C| n⊗Iq, analytic in q ∈ Iq, ∀k ∈ C| n, entire in every complexkj , j = 1, . . . , n, ∀q ∈ Iq, and exhibits a proper asymptotic behavior, forlarge (k, q), in the analyticity regions. If the space domain is semi-bounded,then the analyticity in the Fourier variables kj , j = 1, . . . , n is limited toopen regions of the complex plane, depending on the geometric propertiesof the domain V . We are therefore led to the following definition:

Definition of admissibility. Given a space-time domain D, a function of(k, q) is an admissible Fourier transform for the domain D (an admissibleFTD) iff it possesses the analyticity properties and the asymptotic behaviorcorresponding to that domain.

2.2. THE IBV PROBLEM IN FOURIER SPACE

We find it convenient to rewrite the IBV problem (1.1) in Fourier space.This goal is conveniently achieved using the well-known Green’s formula(identity):

bLa− aLb = div J(x, t), (2.3)

and its integral consequence, the celebrated Green’s integral identity :∫D(bLa− aLb)dxdt =

∫∂DJ(x, t) · νdσ, (2.4)

obtained by integrating (2.3) over the domainD and by using the divergencetheorem. In equation (2.3), L is the formal adjoint of L: L = L(−�,− ∂

∂t),J(x, t) is an (n+1)-dimensional vector field, div is the (n+1)-dimensionaldivergence operator and a(x, t) and b(x, t) are arbitrary functions. In equa-tion (2.4), dσ is the hypersurface element of the boundary and ν is itsoutward unit normal. We remark that, given L, its formal adjoint L andtwo arbitrary functions a and b, an (n+ 1)-dimensional vector field J(x, t)satisfying the Green’s formula (2.3) always exists and can be algorithmicallyfound to be a linear expression of a, b and their partial derivatives of orderup to N − 1, if L is of order N .

The arbitrariness of a and b allows one to extract from (2.3) and (2.4)several important informations on the IBV problem; with the particular


choicea = u(x, t), b = e−i(k·x+qt)/L(ik, iq), (2.5)

where L(ik, iq) is the eigenvalue of the operator L, corresponding to theeigenfunction ei(k·x+qt), the vector field J takes the following form:J = e−i(k·x+qt)J ′(x, t;k, q)/L(ik, iq) and the Green’s integral identity (2.4)gives the FTD of the solution in terms of the FTD of the forcing and (ofappropriate FTs) of all the IBVs:

u(k, q) =1

L(ik, iq) (f(k, q)−∫∂De−i(k·x+qt)J ′(x, t;k, q) · νdσ)

=:N (k, q)L(ik, iq) , (k, q) ∈ A. (2.6)

If the PDE has the following evolutionary form:

L(�, ∂∂t) =

∂

∂t−K(�), (2.7)

which we assume from now on just for the sake of simplicity, then

u(k, q) =f(k, q) + u0(k) + B(k, q)

L(ik, iq) =:N (k, q)L(ik, iq) , (k, q) ∈ A (2.8)

and the linear relation (2.8) makes clear how the different contributions

coming from the equation (the denominator L), from the forcing f , fromthe initial condition u0,

u0(k) =

∫Vdxe−ik·xu0(x), u0(x) := u(x, 0), (2.9)

and from the set of boundary values B separate in Fourier space.Its inverse transform (2.2) gives the corresponding Fourier representa-

tion of the solution:

U(x, t) = u(x, t)χD(x, t) =∫IRn+1

dkdqei(k·x+qt)

(2π)n+1

N (k, q)L(ik, iq) ,

(x, t) ∈ IRn+1. (2.10)

Two sources of problems arise at a first glance of equation (2.6):

i) the RHS of the equation depends on known and unknown BVs;ii) apparently the RHS of the equation is not an admissible FTD.

It is very satisfactory that the analyticity constraints which make ther. h. s. of (2.6) an admissible FTD provide also a number of relations among


the IBVs which are sufficient to express the unknown BVs in terms of knowndata.

2.3. THE ANALYTICITY CONSTRAINTS AND THEIR RESOLUTION

In general, L(ik, iq), the denominator of equation (2.8), is an entire and,most frequently, polynomial function of all its complex variables. Let S bethe manifold in which this entire function is zero:

S = {(k, q) ∈ C| n+1 : L(ik, iq) = 0}. (2.11)

Then the r. h. s. of equation (2.8) provides an admissible FTD of the solutionof the IBV problem under investigation if the numerator N (k, q) of u in(2.8) satisfies in A ∩ S, hereafter called the Singularity Manifold (SM) ofthe IBV problem, the following Zero Residue Condition (ZRC):

N (k, q) = 0, (k, q) ∈ A ∩ S. (2.12)

If the singularity manifold A∩S contains the real axis (which is usuallya part of the boundary of A) and if this singularity is not already takencare of by the ZRC (2.12), then the FT (2.8) of u(x, t) is still admissible,even if it is singular on the real axis, provided one applies the DenominatorRegularization (DR):

L(ik, iq)→ Lreg(ik, iq). (2.13)

In fact, for the PDEs considered here, see (2.7), the zeroes of the denom-inator L(ik, iq) = iq − K(ik) are simple and then one regularizes thedenominator, see (2.13), moving the zeroes of L a bit off the real axis,outside the domain A.

The ZRC plus the DR constitute the main set of Analyticity Constraints(ACs) that must be imposed to the r. h. s. of (2.8) in order to obtain anadmissible FTD of the solution of the IBV problem under investigation.The ZRC (2.12) provides a (linear) relation among the FTs of the forcing,of the initial condition and of all the BVs. Moreover, it is important torealize that the analyticity properties of all these FTs generate, throughthe admissibility argument, a cascade of further analyticity constraints, sothat the analysis goes on until all these conditions are finally met. Thisprocedure defines, in principle, a set of relations (a system of equations)among the IBVs. We then conclude that:

a) The unique solvability of such a system, together with the admissibil-ity of the obtained solution, are equivalent to the study of the uniquesolvability of all the IBV problems associated with (1.1).


b) By solving this system for a set of BVs in terms of the remaining ones,one expresses all quantities in terms of known data and, from equation(2.10), one obtains the Fourier representation of the solution.

In most of the examples considered in this paper, this system of equa-tions is algebraic, with entire coefficients. Therefore, if M is the squaredmatrix of the coefficients of the unknown BVs, the admissibility argumentimposes that the countable set of zeroes of det M :

{qm}m∈ZZ , det M(qm) = 0 (2.14)

lies outside the analyticity domain of an admissible FTD:

qm /∈ Iq, m ∈ ZZ. (2.15)

It turns out that the set (2.14) coincides with the spectrum arising in theeigenfunction expansion approach [9] and coincides also with the restricteddomain in which the EbR method works. These deep connections justifyfor (2.14) the name of spectrum of the IBV problem.

The admissibility argument imposes also that the constructed solutionof such system exhibit the proper asymptotic behavior in the analyticitydomain. It is actually convenient to impose first this asymptotic admissi-bility, the easiest to be checked, which enables one to disregard withouteffort all the IBV problems which are ill-posed because incompatible withasymptotics.

2.4. GENERAL REMARKS

Remark 1. Analyticity vs Causality. It is well-known that there are definiteconnections between the analyticity properties of the FT of the solution ofevolution equations and the causality principle. In our general setting it isstraightforward to show that:The analyticity properties of the FTD of the solution of the IBV problem(1.1) imply the causality principle.Indeed, using the convolution theorem, the inverse FT (2.10) of the r. h. s. ofequation (2.8) (in which all the analyticity constraints have been prelimi-nary imposed) is equivalent to the following Green’s representation of thesolution:

u(x, t) =

t∫0

dt′∫V

dx′GRF (x− x′; t− t′)N (x′, t′), (x, t) ∈ D, (2.16)


where N (x, t)χD(x, t) is the inverse FT (2.2) of N (k, q) and GRF is thecelebrated retarded-fundamental Green’s function of the operator L:

GRF (x, t) =

∫IRn+1

dkdq

(2π)nei(k·x+qt)

Lreg(ik, iq), (2.17)

which satisfies the important property: GRF (x, t) = 0, t < 0, due to theregularization of L(ik, iq). Equation (2.16) is the usual way in which thecausality principle becomes transparent.

Remark 2. Regularization and Fourier representation. As we have alreadywritten, even if the zeroes of the denominator on the real axis are all curedby the ZRC, some regularization must be introduced in the calculationof the Fourier representation (2.10), before splitting N in the sum of thedifferent contributions (each one singular on the real axis) in (2.8) comingfrom the forcing, from the initial condition and from the BVs. The mostconvenient regularization is obviously that in (2.13) and it leads to thefollowing Fourier representation:

U(x, t) = u(x, t)χD(x, t) =∫IRn+1

dkdqei(k·x+qt)

(2π)n+1

f(k, q)

Lreg(ik, iq)

+

∫IRn+1

dkdqei(k·x+qt)

(2π)n+1

u0(k)

Lreg(ik, iq)

+

∫IRn+1

dkdqei(k·x+qt)

(2π)n+1

B(k, q)Lreg(ik, iq)

, (2.18)

where (x, t) ∈ IRn+1.

3. Illustrative Examples

In this section we apply the Analyticity approach to the following classicalequations of the Mathematical Physics in 1+1 dimensions, the Schrodinger,the heat and the linear Korteweg-de Vries (KdV) equations:

∂u

∂t− α∂

2u

∂x2= f, α = i, 1, x ∈ V, t > 0, (3.1)

∂u

∂t− η∂

3u

∂x3= f, η = ±1, x ∈ V, t > 0, (3.2)

prototype examples respectively of second and third order evolutionaryPDEs and basic universal models for the description of dispersive and


diffusive phenomena, where the space domain V is either the segment (0, L)or the semiline (0,∞). Hereafter the BVs will be indicated by

v(j)0 (t) :=

∂ju

∂xj(x, t)|x=0, v

(j)L (t) :=

∂ju

∂xj(x, t)|x=L, j ∈ IN (3.3)

and their Fourier transforms by v(j)0 (q), v

(j)L (q):

v(j)0 (q) :=

∞∫0

dte−iqtv(j)0 (t), v(j)L (q) :=

∞∫0

dte−iqtv(j)L (t). (3.4)

We also apply the method to the study of IBV problems for the multi-dimensional analogue of equation (3.1), for α = i:

∂u

∂t− i� u = f, x ∈ V, t > 0, � :=

n∑j=1

∂2

∂xj2(3.5)

in the n-dimensional quadrant

V = {x : xj ≥ 0, j = 1, . . . , n}. (3.6)

The corresponding BVs will be indicated by:

v(0)0j (xj , t) = u(x, t)|xj=0, v

(1)0j (xj , t) =

∂u∂xj

(x, t)|xj=0 (3.7)

and their FTs by:

v(m)0j (kj , q) =

∞∫0dt

∫Vj

dxje−i(kj ·xj+qt)v(m)

0j (xj , t), m = 0, 1. (3.8)

In equations (3.7)–(3.8) xj = (x1, . . . , xj , . . . , xn) ∈ IRn−1, kj = (k1, . . . ,

kj , . . . , kn) ∈ IRn−1,∫Vjdxj =

∫ L10 dx1 · ·(

∫ Lj0 dxj) · ·

∫ Ln0 dxn, and a line

under a variable indicates that that variable is removed.The application of the Analyticity approach to higher order problems

and to other relevant examples will be presented in [8].

3.1. THE SECOND ORDER PDES

In the case of the PDEs (3.1), equations (2.3) and (2.5) imply:

L = − ∂∂t − α ∂2

∂x2, J =

(ab, α

[a ∂b∂x − b∂a∂x

]),

L(ik, iq) = i(q − iαk2).(3.9)


In addition, if V is the segment (0, L), equation (2.8) yields

u(k, q) =N (k, q)

i(q − iαk2) ,

N (k, q) = f(k, q) + u0(k)− α([v

(1)0 (q) + ikv

(0)0 (q)]

− e−ikL[v(1)L (q) + ikv(0)L (q)]

)(3.10)

and the Fourier representation (2.18) of the solution takes the followingform:

u(x, t) =

∫IR2

dqdk

(2π)2iei(kx+qt)

f(k, q)

q − iαk2 − i0 +∫IR

dk

2πeikx−αk

2tu0(k)

− α∫γ(α)

dk

2πeikx−αk

2t[v(1)0 (iαk2) + ikv

(0)0 (iαk2)]

+ α

∫γ(α)

dk

2πe−ik(x−L)−αk

2t[v(1)L (iαk2)− ikv(0)L (iαk2)] ;

x ∈ (0, L), t > 0 , (3.11)

where the integration path γ(α) = ∂K(α)1 is the counterclockwise oriented

boundary of K(α)1 , and K(i)

1 and K(i)0 are respectively the first and third

quadrant of the complex k-plane, K(1)m = ρπ

4K(i)m , m = 0, 1, where ρπ

4is the

π/4 rotation operator: ρπ4: k → e

iπ4 k (see Figs 1a, b).

The corresponding expressions for the semiline or for the infinite linecases, with rapidly decreasing conditions at ∞, follow immediately from

the ones above, setting v(0)L = v

(1)L = 0 in the semiline case, or setting

v(0)0 = v

(1)0 = v

(0)L = v

(1)L = 0 in the infinite line case.

It is instructive to first apply the Analyticity approach to the simplestcase in which the space domain is the whole space, with rapidly decreasingBVs at x = ±∞.

3.1.1. The whole line V = (−∞,∞)Equation (3.10b) reduces to

N (k, q) = f(k, q) + u0(k) (3.12)

and the admissibility argument imposes that N (k, q)/L be defined in (k, q) ∈A = IR⊗ Iq and be analytic in q ∈ Iq, ∀k ∈ IR. If α = 1, the denominatoris singular for q = ik2, k ∈ IR, outside the definition domain, and noregularization is needed. If, instead, α = i, the denominator is singular


for q = −k2 < 0, on the real negative axis, at the boundary of the an-alyticity domain, and the only analyticity constraint to be fulfilled is theDenominator Regularization (2.13):

L(ik, iq) = i(q − iαk2)→ Lreg(ik, iq) = i(q − iαk2 − i0). (3.13)

The regularization (3.13) is sufficient to make the r. h. s. of the firstequation in (3.10) an admissible FTD, from which we recover the well-known Fourier representation of the solution of equations (3.1):

u(x, t) =

∫IR2

dqdkei(kx+qt)

(2π)2i

f(k, q)

q − iαk2 − i0 +∫IR

dk

2πeikx−αk

2tu0(k) (3.14)

for x ∈ IR, t > 0.

3.1.2. The semiline V = (0,∞)In this case:

N (k, q) = f(k, q) + u0(k) − α[v(1)0 (q) + ikv(0)0 (q)] (3.15)

and admissibility imposes that N/L be defined in A = Ik⊗Iq, be analyticin q ∈ Iq, ∀k ∈ Ik and be analytic in k ∈ Ik, ∀q ∈ Iq. Therefore thesingularity manifolds A∩S(α), corresponding to α = i, 1, are parametrizableeither in terms of k or in terms of q in the following way:

A∩ S(α) = {q = iαk2, k ∈ K(α)0 }

= {k = k(α)0 (q), π ≤ arg q ≤ 2π}, (3.16)

where

k(α)0 (q) =

{iq

12 , α = i

e34πiq

12 , α = 1.

(3.17)

If α = 1, there is no singularity on the real axis and no regularizationis needed (the singularity at k = q = 0 is eliminated by the ZRC). If α = i,there are two singularities for k ∈ IR; that corresponding to k < 0 is curedby the ZRC (2.12), while that corresponding to k > 0 is cured instead bythe regularization (3.13).

The ZRC (2.12) is conveniently parametrized in terms of q in the fol-lowing way:

N (k(α)0 (q), q) = f(k(α)0 (q), q)+ u0(k

(α)0 (q))−α[v(1)0 (q)+ ik

(α)0 (q)v

(0)0 (q)] = 0,

(3.18)


for π ≤ arg q ≤ 2π. It is one equation involving 4 FT ′s which are thereforedependent. If we are interested in solving the Dirichlet and Neumann prob-lems, we use this ZRC to express the unknown BVs in terms of the knownones:

Dirichlet : αv(1)0 (q) = f(k

(α)0 (q), q) + u0(k

(α)0 (q))− αik(α)0 (q)v

(0)0 (q)

Neumann : iαk(α)0 (q)v

(0)0 (q) = f(k

(α)0 (q), q) + u0(k

(α)0 (q))− αv(1)0 (q),

(3.19)for π ≤ arg q ≤ 2π. It is easy to see that the unknown BVs are expressedby (3.19) as admissible FTs which, inserted in (3.11), give the wantedsolution of the Dirichlet and Neumann problems. The Robin problem canbe similarly solved.

We remark that the ZRC (3.18) could also be solved for u0 (using now,for convenience, the variable k):

u0(k) = −f(k, iαk2) + α[v(1)0 (iαk2) + ikv(0)0 (iαk2)], k ∈ K(α)

0 (3.20)

but, in this case, the solution would not be, in general, an admissible FT,since the r. h. s. of (3.20) cannot be extended to the rest of the lower halfk-plane. Even in the special case in which the forcing and the assigned BVswere on a compact support in t, corresponding to entire FTs, the solutionu0(k) would not be admissible, because it would not possess, in general, theproper asymptotics. This means that the (unphysical) problem in which weassign arbitrarily u and its space derivative at x = 0 cannot be treated bythe FT method, unless the above BVs are suitably constrained.

3.1.3. The segment V = (0, L)

Now admissibility implies that N/L be defined in A = C| ⊗Iq, be analytic inq ∈ Iq, ∀k ∈ C| and be analytic in k ∈ C| , ∀q ∈ Iq, with proper asymptoticsfor large |k| and/or |q| in the analyticity regions. Therefore the singularitymanifold on which the ZRC (2.12) is defined is the union of two sectors:

A ∩ S(α) =1⋃

m=0

{q = iαk2, k ∈ K(α)m }

=1⋃

m=0

{k = k(α)m (q) = (−)mk(α)0 (q), π ≤ arg q ≤ 2π}. (3.21)


Fig.1a The SM A ∩ S(α) (α = i) Fig.1b The SM A ∩ S(α) (α = 1)

Both singularities on the real axis are cured by the ZRC and no regu-larization is needed. The regularization (3.13), however, is still introduced,according to the Remark 2 of §2.4, in computing the Fourier representation(2.18) of the solution, and it leads to the r. h. s. of (3.11). The ZRC (2.12),conveniently parametrized using q, consists of the following system of twolinear algebraic equations:

N (k(α)m (q), q) = 0, m = 0, 1 π ≤ arg q ≤ 2π, (3.22)

where N is given by (3.10). Since this system contains four BVs, we expectthat two out of four BVs can be arbitrarily assigned. To establish whichpairs of BVs can be arbitrarily given, one should impose that the corre-sponding solutions of the algebraic system (3.22) define admissible FTs;i.e., the following two conditions must be satisfied.

i) The system must be uniquely solvable for the unknown pair of BVs inits definition domain. More precisely, indicating byM the 2×2 matrixof the coefficients of the unknown BVs, the admissibility conditionimposes that the countable set {qj}j∈IN of zeroes of detM , the spectrumof the IBV problem, lie outside the analyticity domain:

qj /∈ Iq, j ∈ IN. (3.23)

ii) The solution of the system must define admissible Fourier Transforms;in particular, it must exhibit the proper asymptotics in the analyticitydomain.

Studying first the asymptotics of (3.22), one infers without any effort whichpairs of BVs cannot be assigned arbitrarily. The asymptotics of (3.22) implyimmediately that the following expressions:

f(k(α)0 (q), q) + u0(k

(α)0 (q))− α[v(1)0 (q) + ik

(α)0 (q)v

(0)0 (q)], (3.24)

e−ik(α)0 (q)L[f(−k(α)0 (q), q) + u0(−k(α)0 (q))] + α[v

(1)L (q)− ik(α)0 (q)v

(0)L (q)]


are exponentially small for q ∼ ∞ in π ≤ arg q ≤ 2π. Since the asymptoticseries of the admissible FTs appearing in the LHS of equations (3.24) are

inverse power series of q12 , equations (3.24) impose severe constraints on

the involved functions, implying that, if the forcing and the initial data aregiven, then:asymptotic admissibility is compatible with assigning at x = 0 any BV

between (v(0)0 , v

(1)0 ) and, at x = L, any BV between (v

(0)L , v

(1)L ). It is not com-

patible instead with assigning arbitrarily the pairs (v(1)0 , v

(0)0 ) or (v

(1)L , v

(0)L ).

To complete our analysis, we must check if the spectrum associated withthe IBV problems compatible with the asymptotics lie outside the definitiondomain. The analysis is straightforward and produces the following results.

Proposition (the spectrum).Assigning arbitrarily (v(0)0 , v

(0)L ) (the Dirichlet

problem) or (v(1)0 , v

(1)L ) (the Neumann problem), the spectrum is character-

ized by the equation sin(kL) = 0 ⇔ km = πmL , m ∈ ZZ and is given by the

negative eigenvalues {qm}n∈IN , qm = −k2m = −(πmL )2, m ∈ IN , if α = i,and by the purely imaginary eigenvalues qm = ik2m = i(πmL )

2, m ∈ IN , if

α = 1. Assigning instead (v(0)0 , v

(1)L ) or (v

(1)0 , v

(0)L ) (the mixed problems),

the spectrum is characterized by the equation cos(kL) = 0 ⇔ km =πL(2m+1), m ∈ ZZ and is given by the negative eigenvalues {qm}m∈IN , qm =−k2m = −( πL)2(2m + 1)2, m ∈ IN , if α = i, and by the purely imaginaryeigenvalues qm = ik2m = i( πL )

2(2m+ 1)2, m ∈ IN , if α = 1.For α = 1 the spectrum lies outside the analyticity region and the

solutions of the algebraic system (3.22) define directly admissible FTs; ifα = i the solutions of the algebraic system (3.22) define admissible FTsafter moving these singularities a bit off the real q - axis in the UHP,outside the definition domain (again a regularization!). We conclude thatall the IBV problems compatible with admissible asymptotics turn out tobe well-posed:IBV problems for the Schrodinger and heat equations (3.1) are well-posed

assigning at x = 0 any BV among (v(0)0 , v

(1)0 ) and at x = L any BV among

(v(0)L , v

(1)L ).

It is interesting to remark that, if one insisted, instead, in solving an IBV

problem in which the BVs (v(1)0 (q), v

(0)0 (q)) are assigned, the corresponding

algebraic system would be always uniquely solvable (no point spectrumwould arise), but the solution would exhibit an exponential blow up atq ∼ ∞ in the analyticity region, that cannot be accepted. This undesiredblow up could be cured if the assigned BVs were related by the (additional)analyticity constraint:

f(k(α)0 (q), q) + u0(k

(α)0 (q))− α[v(1)0 (q) + ik

(α)0 (q)v

(0)0 (q)] = 0, (3.25)


implying the following admissible solutions of the algebraic system (3.22):

v(1)L (q) = −ik(α)0 (q)v

(0)L (q) = −e

−ik(α)0 (q)L

2α[f(k

(α)0 (q), q) (3.26)

+ f(−k(α)0 (q), q) + u0(k(α)0 (q)) + u0(−k(α)0 (q))− 2αv(1)0 (q)].

The additional analyticity constraint (3.25) is not surprising at all, sinceit is nothing but the ZRC of the semiline problem. Similarly, assigning the

right boundary conditions (v(0)L , v

(1)L ), the unknowns v

(0)0 and v

(1)0 would

exhibit again an exponential blow up which cannot be accepted; an admis-sible asymptotics would be guaranteed now by the (additional) analyticityconstraint:

e−ik(α)0 (q)L[f(−k(α)0 (q), q) + u0(−k(α)0 (q))] + α[v

(1)L (q)− ik(α)0 (q)v

(0)L (q)] = 0,

(3.27)implying the following solution of the algebraic system:

v(1)0 (q) = ik

(α)0 (q)v

(0)0 (q) =

1

2α[f(k

(α)0 (q), q) + f(−k(α)0 (q), q) (3.28)

+ u0(k(α)0 (q)) + u0(−k(α)0 (q)) + 2αe−ik

(α)0 (q)Lv

(1)L (q)].

3.1.4. The periodic problem

If we assume L-periodicity of u and ux, then v(1)0 = v

(1)L =: v(1), v

(0)0 =

v(0)L =: v(0) and the algebraic system (3.22) consists now of two equationsfor two BVs, which have to be treated therefore as unknowns. The solutionsof this system read:

v(1)(q) = 12α

(f(k

(α)0 (q),q)+u0(k

(α)0 (q))

1−e−ik(α)0

(q)L+f(−k(α)0 (q),q)+u0(−k(α)0 (q))

1−eik(α)0

(q)L

),

v(0)(q) = 1

2iαk(α)0 (q)

(f(k

(α)0 (q),q)+u0(k

(α)0 (q))

1−e−ik(α)0

(q)L− f(−k(α)0 (q),q)+u0(−k(α)0 (q))

1−eik(α)0

(q)L

).

(3.29)

They satisfy asymptotic admissibility and the spectrum, characterized bythe equation 1−e±ikL = 0, (⇒ kn =

2πL n, n ∈ ZZ), is given by qn = −k2n =

−(2πL )2n2, n ∈ IN , for α = i, and by qn = ik2n = i(2πL )2n2, n ∈ IN , for

α = 1; therefore the usual regularization is needed again in the Schrodingercase. We conclude that the periodic problem for equations (3.1), in whichone imposes the L- periodicity of u and ux, is well-posed and no BV can beassigned arbitrarily.

RemarkWe remark that the Fourier transforms of the unknown boundaryfunctions exhibit generically a branch point at q = 0, due to the well-known


slow decay as t → ∞ of the solutions of the dispersive evolution equationunder investigation.

The above procedure generalizes with no difficulties to higher orderproblems. In the following we concentrate on a third order problem only.

3.2. THE LINEAR KDV EQUATION

In this section we investigate IBV problems for 3rd order operators, illus-trating the method on the simplest possible example (3.2).

Since the group velocity vg = 3ηk2 of the associated wave packet ispositive (negative ) for η positive (negative), we have the following expec-tations. In the semiline case, one should be able to assign at x = 0 moreBVs for positive η than for negative η. In the segment case, for η positiveone can assign arbitrarily more BVs at x = 0 than at x = L (and vice versafor η negative). The precise indication of “how many” and “which” BVscan be assigned in order to have a well-posed IBV problem follows again ina straightforward way from the Analyticity approach.

Equations (2.3) and (2.5) imply:

L = −L, J =(ab,−η

[b∂2a

∂x2− ∂b∂x

∂a

∂x+∂2b

∂x2a]),

L(ik, iq) = i(q + ηk3).(3.30)

In addition, if V is the segment (0, L), equation (2.8) yields

u(k, q) = −i N (k,q)q+ηk3 ,

N (k, q) = f(k, q) + u0(k) − η([v

(2)0 (q) + ikv

(1)0 (q)− k2v(0)0 (q)]

− e−ikL[v(2)L (q) + ikv(1)L (q)− k2v(0)L (q)]

) (3.31)

and the Fourier representation (2.18) of the solution takes the followingform:

u(x, t) =

∫IR2

dqdk

(2π)2iei(kx+qt)

f(k, q)

q + ηk3 − i0 +∫IR

dk

2πei(kx−ηk

3t)u0(k)

+ η

∫γ

dk

2πeik

3t2∑

m=0

(−iηk)m[v(2−m)0 (k3)E(m)(k, ηx) (3.32)

− v(2−m)L (k3)E(m)(k, η(x − L))], x ∈ (0, L), t > 0,

where γ is the counterclockwise oriented boundary of the sector K(−)0 , γ =

∂K(−)0 (see Fig.2a), and

E(m)(k, x) := H(x)(ρm+1e−iρ1kx + ρ2−me−iρ2kx)−H(−x)e−ikx, (3.33)


ρm being the cubic roots of unity:

ρm = e2πi3m. (3.34)

3.2.1. The segment V = (0, L)

Now N/L must be defined in A = C| ⊗ Iq, analytic in q ∈ Iq, ∀k ∈ C| andanalytic in k ∈ C| , ∀q ∈ Iq, with proper asymptotics for large |k| and/or|q| in the analyticity regions. Therefore the singularity manifold A ∩ S(η),corresponding to η = ±1, are given by (see Figs 2a,b):

A ∩ S(η) =2⋃

m=0

{q = −ηk3, k ∈ K(η)m }

=2⋃

m=0

{k = k(η)m (q), π ≤ arg q ≤ 2π} (3.35)

where k(η)m (q) = −ηρmq

13 and the three sectors K(η)

m are

K(−)m = {k : π

3(2m+ 1) ≤ arg k ≤ π

3(2m+ 2)}, (3.36)

K(+)m = ρπK(−)

m , m = 0, 1, 2, and ρπ is the involution ρπ : k → −k.

Fig.2a The SM A∩ S(η) (η = −1) Fig.2b The SM A ∩ S(η) (η = 1)

The ZRC (2.12) consists of the following three equations:

N (k(η)m (q), q) = 0, m = 0, 1, 2, π ≤ arg q ≤ 2π. (3.37)

For q ∈ IR there is one singularity on the real k - axis, which is cured byone of the three equations (3.37) and no denominator regularization is thenneeded. The regularization (3.13), however, is still introduced, according to


the Remark 2 of §2.2.2, in writing the Fourier representation (3.32) of thesolution.

The three3 algebraic equations (3.37) contain six BVs; therefore weexpect to be allowed to assign independently only three BVs. As before, aquick asymptotic estimate selects the sets of three BVs which can be as-signed independently, compatibly with asymptotic admissibility. The asymp-totics of equations (3.37) imply that the following expressions, respectively,for η = −1:

eiq13 L[f(q

13 , q) + u0(q

13 )]− [v(2)L (q) + iq

13 v

(1)L (q)− q 23 v(0)L (q)],

f(ρ1q13 , q) + u0(ρ1q

13 ) + [v

(2)0 (q) + iρ1q

13 v

(1)0 (q)− ρ2q

23 v

(0)0 (q)],

f(ρ2q13 , q) + u0(ρ2q

13 ) + [v

(2)0 (q) + iρ2q

13 v

(1)0 (q)− ρ1q

23 v

(0)0 (q)],

(3.38)

and for η = 1:

f(−q 13 , q) + u0(−q13 )− [v(2)0 (q)− iq 13 v(1)0 (q)− q 23 v(0)0 (q)],

e−iρ1q13 L[f(−ρ1q

13 , q) + u0(−ρ1q

13 )]

+ [v(2)L (q)− iρ1q

13 v

(1)L (q)− ρ2q

23 v

(0)L (q)],

e−iρ2q13 L[f(−ρ2q

13 , q) + u0(−ρ2q

13 )]

+ [v(2)L (q)− iρ2q

13 v

(1)L (q)− ρ1q

23 v

(0)L (q)],

(3.39)

are exponentially small for q ∼ ∞ in π ≤ arg q ≤ 2π. Therefore, reasoningas before, we see that:

i) for η = −1, a necessary and sufficient condition to obtain FTs withadmissible asymptotics is to assign at x = 0 any one BV among

v(0)0 , v

(1)0 , v

(2)0 (consequence of equations (3.38b,c)) and, at x = L, any

two BVs among v(0)L , v

(1)L , v

(2)L (consequence of equation (3.38a));

ii) for η = 1, a necessary and sufficient condition to obtain FTs withadmissible asymptotics is to assign at x = 0 any two BVs among

v(0)0 , v

(1)0 , v

(2)0 (consequence of equation (3.39a)) and, at x = L, any one

BV among v(0)L , v

(1)L , v

(2)L (consequence of equations (3.39a,b)).

Again, to complete our investigation, we must check if the spectrum associ-ated with the above IBV problems selected by the asymptotic admissibility,lie entirely outside the analyticity domain of an admissible FT. It is easyto prove that it is indeed the case.Proposition (the spectrum of the IBV problem). Consider any IBV prob-lem on the segment for equation (3.2) compatible with the asymptotic admis-sibility established above; i.e., in which, for η = −1, one assigns arbitrarily

at x = 0 any BV among v(0)0 , v

(1)0 , v

(2)0 and any two BVs at x = L among

v(0)L , v

(1)L , v

(2)L , and in which, for η = 1, one assigns arbitrarily at x = 0 any


two BVs among v(0)0 , v

(1)0 , v

(2)0 and any BV at x = L among v

(0)L , v

(1)L , v

(2)L .

For η = −1, let v(n)0 be the given BV at x = 0 and v(m)L be the unknown BV

at x = L while, for η = 1, let v(n)0 be the unknown BV at x = 0 and v

(m)L be

the given BV at x = L. Then the corresponding spectrum is characterizedby the following equation:

Δ(η(m−n))(k) = 0, (3.40)

where:Δ(j)(k) := e−ikL + ρj1e

−ρ1ikL + ρj2e−ρ2ikL. (3.41)

The proof is tedious but straightforward and makes essential use of the well-known algebra of the roots of unity, which implies also that all the aboveIBV problems lead only to three (similar) purely imaginary discrete spectra

{k(j)n }n∈IN , characterized by the three equations Δ(j)(k) = 0, j = 0, 1, 2.More precisely:

1) the spectrum characterized by equation Δ(0)(k) = 0 is given by:

k(0)n = −i(ζ(0)n /L), n ∈ IN+ : e

− 32ζ(0)n

2 = − cos(√32 ζ

(0)n )

(ζ(0)n ∼ π√

3(2n − 1), n ≥ 1).

(3.42)

2) The spectrum characterized by equation Δ(1)(ζ) = 0 is:

k(1)n = −i(ζ(1)n /L), n ∈ IN : e−

32 ζ

(1)n

2 = cos(√32 ζ

(1)n + π

3 ),

(ζ(1)0 = 0, ζ

(1)n ∼ 2π√

3(n− 5

6), n ≥ 2).(3.43)

3) The spectrum characterized by equation Δ(2)(ζ) = 0 is:

k(2)n = −i(ζ(2)n /L), n ∈ IN : e−

32ζ(2)n

2 = cos(√32 ζ

(2)n − π

3 ),

(ζ(2)0 = 0, ζ

(2)n ∼ 2π√

3(n− 1

6), n ≥ 1).(3.44)

We conclude that all the three discrete spectra

{q(j)n }n∈IN , q(j)n = k(j)n3= i

(ζ(j)n

L

)3

, j = 0, 1, 2, (3.45)

associated with the above IBV problems lie on the positive imaginary axisof the complex q plane, outside the analyticity domain of an admissible FT.Therefore:IBV problems for equation (3.2) on the segment (0, L) are well-posed iff:


i) for η = −1, one assigns at x = 0 any one BV among v(0)0 , v

(1)0 , v

(2)0 and

at x = L any two BVs among v(0)L , v

(1)L , v

(2)L ;

ii) for η = 1, one assigns at x = 0 any two BVs among v(0)0 , v

(1)0 , v

(2)0 and

any one BV at x = L among v(0)L , v

(1)L , v

(2)L .

3.2.2. The periodic problem

If we assume L-periodicity of u, ux and uxx, then v(j)0 = v

(j)L , j = 0, 1, 2, the

algebraic system (3.37) consists now of three equations for three BVs, whichhave to be treated then as unknowns. The solution of this system satisfyasymptotic admissibility and the spectrum, characterized by the equations1−e−iρjk = 0, j = 0, 1, 2 (⇒ kn =

2πL ρ

−1j n, n ∈ ZZ, j = 0, 1, 2), is given by

the real numbers qn = −ηk3n = −η(2πL )3n3, n ∈ ZZ and must be regularizesin the usual way. We conclude that:the periodic problem for the linear KdV equation (3.2), in which one im-poses L-periodicity to u, ux and uxx, is well-posed and no BV can beassigned.

3.2.3. The semiline V = (0,∞)Taking the limit L → ∞ of the results of §3.2.1 we immediately obtainthe results on the semiline. In this case, the singularity manifolds are therestrictions of the above ones to the lower half k plane. No spectrum arisesand the asymptotic admissibility implies that:IBV problems for equation (3.2) on the semiline (0,∞) are well-posed iff,

for η = −1, one assigns at x = 0 any one BV among (v(0)0 , v

(1)0 , v

(2)0 ) and,

for η = 1, one assigns at x = 0 any two BVs among (v(0)0 , v

(1)0 , v

(2)0 ).

We remark that, in the cases treated so far, the spectra of all the IBVproblems compatible with asymptotic admissibility lie always outside theanalyticity domain. We do not have, however, a general argument exclud-ing the situation in which part of the spectrum lie inside. Therefore thecomplete characterization of the spectrum, the only part of the method inwhich some technicality is involved, seems to be unavoidable and makes itdifficult to prove general results for operators of arbitrary order.

The Analyticity approach applies nicely also to an arbitrary numberof dimensions and next section is devoted to an illustration of it. Theapplication of the method to higher order problems and to other relevantexamples will be presented in [8].


3.3. MULTIDIMENSIONAL SCHRODINGER EQUATION

In this section we study the Dirichlet and Neumann problems for theSchrodinger equation (3.5) in the n-dimensional quadrant (3.6). Then:

L = − ∂∂t − i�, J = (ab, i(a� b− b� a),

L(ik, iq) = i(q + k2),(3.46)

where k2 = k · k. Equations (2.6) and (3.46) give the following expressionof the Fourier transform of the solution in terms of the Fourier transformsof the forcing and of all the IBVs:

u(k, q) = N (k,q)i(q+k2)

,

N (k, q) := f(k, q) + u0(k)− in∑j=1[v

(1)0j (kj , q) + ikj v

(0)0j (kj , q)].

(3.47)

The Fourier representation (2.18) of the solution reads:

u(x, t) =∫IRn+1

dqdk(2π)n+1ie

i(k·x+qt) f(k,q)q+k2−i0 +

∫IRn

dk(2π)n e

i(k·x−k2t)u0(k)

+n∑j=1

∫IRn−1

dkj(2π)n−1

∫γ(i)

dkj2πi{ei(k·x−k

2t)[v(1)0j (kj ,−k2) + ikj v

(0)0j (kj ,−k2)],

(3.48)

where dkj = dk1 · · · dkj · · · dkn.In view of the distinguished parity properties of the Fourier transforms

in (3.47), we shall make an extensive use of the parity operators:

Δ± =n∏l=1

(1±σl), Δ(j)± =

n∏l=1l�=j

(1±σl), (3.49)

where σj is the involution σj : kj → − kj .In this multidimensional case, the FT of the solution is defined in A =

Ik1⊗· · ·⊗Ikn⊗Iq, analytic for q ∈ Iq, ∀k ∈ Ik1⊗· · ·⊗Ikn and in kj ∈ Ikj ,∀kj ∈ Ik1⊗· · ·⊗ Ikj⊗· · ·⊗ Ikn and ∀q ∈ Iq. We find it convenient to studythe ZRC in the n different regions Q−j ⊂ A ∩ S, j = 1, . . . , n defined by:

Q−j := {(k, q) ∈ C| n+1 : kj ∈ IRn−1, π ≤ arg q ≤ 2π, kj = χj(kj , q)},

χj(kj , q) := i(q + kj · kj)12 ∈ Ik, j = 1, . . . , n. (3.50)

Therefore the starting point of the analysis is the set of n equations

N (k, q)|kj=χj = 0, kj ∈ IRn−1, π ≤ arg q ≤ 2π, j = 1, . . . , n.(3.51)


Dirichlet problem. The parity properties in k of the BV terms imply that the

application of the parity operator Δ(j)− defined in (3.49) to the jth equation

(3.51) eliminates all the v(1)0 s except v

(1)0j :

Δ(j)− v

(1)0j (kj , q) = −Δ(j)

− (W (k, q)|kj=χj), j = 1, . . . , n,

W (k, q) := if(k, q) + iu0(k) + in∑j=1kj v

(0)0j (kj , q)

(3.52)

and the analyticity properties of the functions v(1)0 allow one to express

them in terms of known quantities:

v(1)0j (kj , q) = P(j)Δ

(j)− v

(1)0j (kj , q) = −P(j)Δ

(j)− (W (k, q)|kj=κj ), (3.53)

for j = 1, . . . , n, applying the lower half plane analyticity projectors in allthe k-variables (except kj):

P(j) =n∏

m=1m�=j

Pm, Pm = − 1

2πi

∫I

Rdk′m

k′m − (km − i0). (3.54)

Equations (3.53) summarize all the analyticity informations contained inthe ZRC, allow one to express the unknown BVs in terms of given dataand, via (3.48), to solve the Dirichlet problem.Neumann problem. Similar considerations can be made in solving the Neu-mann BV problem. In this case:

iχj(kj , q)Δ(j)+ v

(0)0j (kj , q) = −Δ

(j)+ (V (k, q)|kj=χj), j = 1, . . . , n,

V (k, q) := if(k, q) + iu0(k) +n∑j=1

v(1)0j (kj , q)

(3.55)

and

v(0)0j (kj , q) = iP(j)

(1

χj(kj ,q)Δ

(j)+ (V (k, q)|kj=χj)

), j = 1, . . . , n. (3.56)

In this multidimensional context, for the presence of the analyticityprojectors, the unknown BVs in Fourier space turn out to be nonlocalexpressions of the given data. It is however possible to show that, due tothe analyticity properties of the involved FTs, it is not really necessary toapply the above analyticity projectors to construct the unknown BVs andthe solution u(x, t) in configuration space. The strategy to avoid unpleasantnonlocalities is outlined in the next section and leads to a Fourier repre-sentation of the solution already obtained in [6] using the EbR approach.Therefore this strategy is also the way to establish the connection betweenthe Analyticity and the EbR approaches.


4. Connections between the Analyticity and the EbR approaches

Dirichlet problem. We first remark that the unknown BVs can be con-structed directly in terms of known data from the r. h. s. of (3.52b):

v(1)0j (xj , t) = −

∫IRn

dkjdq

(2π)nei(kj ·xj+qt)Δ(j)

− (W (k, q)|kj=χj), (4.1)

for t > 0, xk ≥ 0, k �= j. Indeed, from the analyticity properties of v(1)0j we

know that its inverse FTs (2.2) is zero outside the domain of definition inconfiguration space (i.e., for xk < 0, k �= j); this implies the formula∫

IRndkjdqe

i(kj ·xj+qt)[v(1)0j (kj , q)−Δ(j)− v

(1)0j (kj , q)] = 0 (4.2)

for t > 0, xj > 0, j = 1, . . . , n and, through (3.52a), equation (4.1).Also the solution u(x, t) can be reconstructed without going through the

nonlocalities associated with the analyticity projectors. Indeed it is possibleto show that the following relation holds true:

n∑j=1

v(1)0j (kj , q) ≡ (Δ− − 1)W (k, q), (4.3)

where the equivalence A(k, q) ≡ B(k, q) means that the FTs A(k, q) andB(k, q) are equal under the following Fourier integral projector:∫

IRn+1

dkdqei(k·x+qt)

q + k2 − i0[A(k, q) − B(k, q)] = 0, (x, t) ∈ D. (4.4)

The equivalence (4.3) and equation (3.52) imply

N (k, q) ≡ Δ−W (k, q) (4.5)

and the following spectral representation of the solution in terms of knowndata:

u(x, t) =

∫Rn+1

dqdk

(2π)n+1ei(k·x+qt)

Δ−f(k, q)q + k2 − i0

+

∫Rn

dk

(2π)nei(k·x−k

2t)Δ−u0(k) (4.6)

+n∑j=1

∫IRn−1

dkj(2π)n−1

(

∫γ(i)

dkjπei(k·x−k

2t)kjΔ(j)− v

(0)0j (kj ,−k2),

(x, t) ∈ D,


already obtained in [6] using the EbR approach.The proof of (4.3) is based on the important fact that all the admissible

Fourier transforms A(k, q) under consideration satisfy the equivalence

σjA(k, q) ≡ A(k, q)|kj=χj , j = 1, . . . , n (4.7)

and goes as follows. For n = 2, the 2 ZRCs (3.51) and their consequences(3.52) yield the 4 equivalence relations:

v(1)01 (k2, q) + σ1v

(1)02 (k1, q) ≡ −σ1W (k, q),

σ2v(1)01 (k2, q) + v

(1)02 (k1, q) ≡ −σ2W (k, q),

(1− σ2)v(1)01 (k2, q) ≡ −σ1(1− σ2)W (k, q),

(1− σ1)v(1)02 (k1, q) ≡ −σ2(1− σ1)W (k, q)

(4.8)

and their sum is exactly equation (4.3). To generalize this result to the caseof an arbitrary n, consider the n ZRCs (3.51) and all their consequences,obtained applying systematically parity operators characterized by differentindexes:

v(1)0j (kj , q) + σj

∑l �=jv(1)0l (kl, q) ≡ − σjW (k, q), j = 1, . . . , n

(1− σi)v(1)0j (kj , q) + σj(1− σi)∑l �=jv(1)0l (kl, q) ≡ −σj(1− σi)W (k, q) ,

i �= j, j = 1, . . . , n ;. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Δ(j)− v

(1)0j (kj , q) ≡ −σjΔ

(j)− W (k, q), j = 1, . . . , n.

(4.9)

The sum of all these equations with weights 1/(n−1m

)(m is the number of

parity operators appearing in the equation) yields the result (4.3).Neumann problem. Similar considerations can be made in the case of theNeumann IBV problem. Now the unknown BVs are recovered via:

v(0)0j (xj , t) = −

∫IRn

dkjdq

(2π)nei(kj ·xj+qt)(Δ(j)

+ V (k, q)|kj=χj ) (4.10)

for t > 0, xk ≥ 0, k �= j, and the spectral representation of the solutionreads:

u(x, t) = −∫IRn+1

dqdkei(k·x+qt)

(2π)n+1iΔ+f(k,q)q+k2−i0 +

∫IRn

dkei(k·x−k2t)

(2π)n Δ+u0(k)

−n∑j=1

∫IRn−1

dkj(2π)n−1 (

∫γ(i)

dkjei(k·x−k2t)

πi Δ(j)+ v

(1)0j (kj ,−k2), (x, t) ∈ D;

(4.11)

a formula already derived in [6] using the EbR approach.


Acknowledgments

The present work has been carried out during several visits and meet-ings. We gratefully acknowledge the financial contributions provided by theRFBR Grant 01-01-00929, the INTAS Grant 99-1782 and by the followingInstitutions: the University of Rome “La Sapienza” (Italy), the IstitutoNazionale di Fisica Nucleare (Sezione di Roma), the Landau Institute forTheoretical Physics, Moscow (Russia) and the Isaac Newton Institute,Cambridge (UK), within the programme “Integrable Systems”.

References

1. A. S. Fokas, Proc. Roy. Soc. Lond. A, 53, 1411 (1997).2. A. S. Fokas, J. Math. Phys. 41, 4188 (2000).3. A. S. Fokas, Proc. Roy. Soc. Lond. A, 457, 371 (2001).4. A. S. Fokas and B. Pelloni, Math. Proc. Camb. Phil. Soc., 131, 521 (2001).5. B. Pelloni, “On the well-posedness of boundary value problems for integrable evo-

lution equations on a finite interval”, in the Proceedings of NEEDS 2001; editorsA. Mikhailov and P. M. Santini. To be published in Theor. Math. Phys., 2002.

6. A. Degasperis, S. V. Manakov and P. M. Santini, “Initial-boundary value problemsfor linear and soliton PDEs”, Theor. Math. Phys., 133, 1472 (2002).

7. A. S. Fokas, “A new transform method for evolution PDEs”, preprint 2002.8. A. Degasperis, S. V. Manakov and P. M. Santini, “Initial-boundary value problems

for integrable PDEs: spectral methods of solution”; in preparation.9. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Mc Graw-Hill, New

York (1953).

QUASI-EXACTLY SOLVABLE BOSE SYSTEMS

S.N. DOLYA ([email protected])B. Verkin Institute for Low Temperature Physics andEngineering, 47 Lenin Prospekt, Kharkov 61164, Ukraine

O.B. ZASLAVSKII ([email protected])Department of Mechanics and Mathematics, KharkovV.N. Karazin’s National University, Svoboda Sq. 4,Kharkov 61077, Ukraine

Abstract. We extend the notion of quasi-exactly solvable (QES) models to Bose systems.We obtain conditions under which algebraization of the part of the spectrum occurs. Insome particular cases simple exact expressions for several energy levels of an anharmonicBose oscillator are obtained explicitly. The corresponding results do not exploit perturba-tion theory and include strong coupling regime. A generic Hamiltonian under discussioncannot, in contrast to QES potential models, be expressed as a polynomial in generatorsof sl2 algebra. The suggested approach is extendable to many-particle Bose systems withinteraction.

1. Introduction

The conception of quasi-exactly solvable (QES) systems, discovered in the1980s [1]–[12], received in recent years much attention both from the view-point of physical applications and their inner mathematical beauty. Inturned out that in quantum mechanics there exists a peculiar class ofobjects that occupy an intermediate place between exactly solvable andnon-solvable models in the sense that in an infinite Hilbert space a finitepart of a spectrum is singled out within which eigenvectors and eigenvaluescan be found from an algebraic equation of a finite degree—in other words,a partial algebraization of the spectrum occurs. For one-dimensional QESmodels corresponding QES Hamiltonians possess hidden group structurebased on sl(2, R) algebra. Thus, they have direct physical meaning, beingrelated to quantum spin systems [13].

Meanwhile, the notion of QES systems is not constrained by potentialmodels and can have nontrivial meaning for any kind of infinite-dimensionalsystems. In the first place, it concerns Bose Hamiltonians whose physical

105


106 S.N. DOLYA AND O.B. ZASLAVSKII

importance is beyond doubt. Here one should distinguish two cases. First,it turns out that some systems of two interacting particles or quasi-particleswith Bose operators of creation and annihilation a, a+ and b, b+ can bemapped on the problem for a particle moving in a certain type of one-dimensional potentials and, remarkably, these potentials belong just to theQES type [14]–[16]. In particular, such a type of Hamiltonians is widelyspread in quantum optics and physics of magnetism [13]. The aforemen-tioned mapping works only for a special class of Bose Hamiltonians whichpossess an integral of motion. Then the procedure is performed in threesteps: (i) the whole Hilbert space splits in a natural way into different pieceswith respect to the values of an integral of motion, (ii) in each piece theSchrodinger equation takes a finite-difference form, (iii) it is transformedinto the differential equation by means of introducing a generating function.In so doing, the integral of motion under discussion represents a linearcombination of numbers of particles a+a and b+b.

The second kind of Bose systems looks much more usual—it is simplysome polynomial with respect to Bose operators of creation and annihila-tion of one particle. The fact that only one pair a, a+ enters Hamiltonian,deprives us, by contrast with the first case, of the possibility to constructa simple integral of motion—in this sense the eigenvalue problem becomesmore complicated. In general, the solutions of the Schrodinger equationcontain infinite numbers of quasi-particles and only approximate or nu-merical methods can be applied to such systems. However, it turns outthat, if the coefficients at different powers of a, a+ are selected in a properway, in some cases a finite-dimensional closed subspace is singled out andalgebraization of the spectrum occurs similar to what happens in ”usual”QES potential models or differential equations. In so doing, the eigenvectorsbelonging to the subspace under discussion, can be expressed as a finite lin-ear combination of eigenvectors of an harmonic oscillator and, thus, containa finite number of quasi-particles. The relevant basis functions that composea finite-dimensional subspace, look very much unlike the wave functions ofa harmonic oscillator. As a result, we obtain QES models with an infinitenumber of quasi-particles in this finite-dimensional subspace.

2. Anharmonic Bose-oscillator

Consider Hamiltonian

H = H0 + V, H0 =p0∑p=1

εp(a+a)p,

V =s0∑s=0

As[(a+a)sa2 + (a+)2(a+a)s]. (2.1)

QUASI-EXACTLY SOLVABLE BOSE SYSTEMS 107

Throughout the paper we assume that all the coefficients of the Hamiltonianare real. For Hamiltonian (2.1) to have a well-defined ground state, oneshould take p0 > s0 + 2 independently of the relations between coefficientsor p0 = s0 + 2 provided εp0 ≥ 2As0 . In the x-representation we obtain

Hx =p0∑p=1

εp

(xd

dx

)p+

s0∑s=0

As

[(xd

dx

)s d2dx2

+ x2(xd

dx

)s]. (2.2)

We are interested in the solutions of Schrodinger equation of the type |ψ〉 =∑Nn=0 bn |n〉, where |n〉 is the state with n particles: a+a |n〉 = n |n〉. For

Hamiltonian (2.2) subspaces with even and odd are not mixed. Therefore,it makes sense to consider them separately. In x representation a → d

dx ,

a+ → x the wave function of even states Φ =∑l=0 alΦl, Φl ≡ x2l.

It follows from (2.2) that

HxΦl = αlΦl+1 + βlΦl−1 + γlΦl , l = 0 , 1, . . . , L , (2.3)

αl =s0∑s=0

As(2l)s ,

βl =s0∑s=0

As2l(2l − 1)(2l − 2)s ,

γl =p0∑p=1

εp(2l)p .

We are interested in the possibility of the existence of the invariant basicF2L = {1, x2, x4, . . . , x2L}. The condition of cut off at l = L reads: αL = 0.

For odd states the invariant basic F2M+1 = {x, x3, . . . , x2M+1}, Φm =x2m+1 and

HxΦm = αmΦm+1 + βmΦm−1 + γmΦm , m = 0, 1, . . . ,M , (2.4)

αm =s0∑s=0

As(2m+ 1)s ,

βm =s0∑s=0

As(2m+ 1)2m(2m − 1)s ,

γm =p0∑p=1

εp(2m+ 1)p .

The subspace with m ≤M is invariant with respect to the action of Hprovided αM = 0.


The procedure described above is, in fact, nothing else than the Boseversion of quasi-exactly solvable (QES) models, applied to an anharmonicoscillator. Now we would like to point out why for the case under con-sideration Turbiner’s theorem, in general, does not hold, so our formulascannot considered as particular cases of its realization. The point is thatTurbiner’s theorem implies that the space FN = {1, x, x2, . . . , xN} of allpolynomials of degree at most N is invariant with respect to J i. Meanwhile,in our case, only subset F2N (for even states) or F2M+1 (for odd ones) isinvariant, whereas the set FN is not. Only in particular cases, when bothconditions αL = 0 (for even states) and αM = 0 (for odd states) are satisfiedsimultaneously, the Hamiltonian becomes an algebraic combination of J i.

In contrast to [11], where differential equations were the object of re-search, in our paper the coordinate-momentum representation, in which theoperator a becomes differential, is used as an useful device at an intermedi-ate stage only. In principle, one could rely directly on the known formulasof the action of operators a, a+ on states with a definite number of particleswithout resorting to the x-representation.

Consider, as an example, the Hamiltonian (2.1) with s0 = 2, whoseoff-diagonal part reads

V = A0(a2 + a+2) +A1[(a

+a)a2 + a+2(a+a)]

+A2[(a+a)2a2 + a+2(a+a)2] . (2.5)

Now

αl = A0 + 2lA1 + (2l)2A2 , (2.6)

βl = 2l(2l − 1)[A0 + (2l − 2)A1 + (2l − 2)2A2] .

αm = A0 + (2m+ 1)A1 + (2m+ 1)2A2 , (2.7)

βm = (2m+ 1)2m[A0 + (2m− 1)A1 + (2m− 1)2A2] .

First, consider even states. In the simplest nontrivial particular case theinvariant subspace is two-dimensional, L = 1. Then

α1 = A0 + 2A1 + 4A2 = 0 , (2.8)

Φ = a0Φ0 + a1Φ1 and it follows from the Schrodinger equation HΦ = EΦthat −Ea0 + β1a1 = 0, α0a0 + (γ1 − E)a1 = 0. Taking also into account(2.8), we obtain:

E =γ12±√γ214+ 8(A1 + 2A2)2 (2.9)


In a similar way, one gets for the three-dimensional subspace (L = 2):

E3 − (γ1 + γ2)E2 + [γ1γ2 − 16(5A21 + 52A1A2 + 140A

22)]E

+ 32γ2(A1 + 4A2)2 = 0 . (2.10)

If A1 = A2 = 0, the low-lying energy levels of an harmonic oscillator arereproduced from (2.9), (2.10). Equation (2.10) can be easily solved in theparticular case A1 = −4A2:

E = 0 ,γ1 + γ22

±√(γ1 − γ2)2

4+ 192A2

2 .

For odd states in the simplest nontrivial case M = 1 we have

α1 ≡ A0 + 3A1 + 9A2 = 0 ,

α0 = A0 +A1 +A2 , β1 = 6α0 ,

E =γ0 + γ12

±√(γ0 + γ1)2

4+ α0β1

=γ0 + γ12

±√(γ0 + γ1)2

4+ 24(A1 + 4A2)2 . (2.11)

The conditions αL = 0 and αM = 0 are different, so in general theinvariant subspace exists only for even or only for odd states. However, itmay happen that both conditions are fulfilled. Thus, for L = 1 = M thecompatibility of (2.8) and (2.11) demands A1 = −5A2, A0 = 6A2. Then wehave the simple explicit solutions for 4 levels of Hamiltonian (2.1):

E =γ0 + γ12

±√(γ0 + γ1)2

4+ 24A2

2 ,γ12±√γ214+ 72A2

2 .

A similar procedure can be repeated for odd states.

3. Generalization

Consider the operator which is the even polynomial of the forth degreewith respect to Bose operators of creation a+ and annihilation a. It can bewritten in the form

H = a++K2+ + a−−K

2− + a00K

20 + a0−K0K− + a+0K+K0

+ a0K0 + a−K− + a+K+ , (3.1)

where


K0 =1

2

(a+a+

1

2

),K− =

a2

2,K+ =

a+2

2, (3.2)

[K0, K±] = ±K± , [K+,K−] = −2K0 . (3.3)

The Casimir operator is C = K20 − 1

2(K+K− +K−K+) ≡ − 316 .

We will use the coherent state representation in which

a→ ∂

∂z, a+ → z (3.4)

After substitution into (3.1) Hamiltonian H(a+, a) becomes a differentialoperator H(z, ∂∂z ). In our previous article [17] we discussed Bose systemsthat possess an invariant subspace of the form F = span{zn} or span{z2n}.The first natural step towards generalization consists in considering sub-spaces (with N fixed )

F = span{un} , un = z2nu , n = 0, 1, 2, . . . , N , (3.5)

for which the following procedure should be realized. (i) The action ofoperators of Ki on the functions un should lead to the linear combinationsof functions from the same set {un}, (ii) by the selection of appropriatecoefficients in (3.1), we achieve the subspace F to be closed under the actionof Hamiltonian H. We would like to stress that the condition (i) does notforbid un with n > N to appear in terms like Kiun but the condition (ii)rules out such functions from Hun (recall that we consider Hamiltonianswhich are quadratic-linear combinations of Ki).

It is seen from (3.2), (3.4) that the operators Ki contain z and∂∂z .

Therefore, it is convenient to assume that differentiation of u(z) gives riseto u up to the factor that contains powers of z. The corresponding choice isnot unique. In the present article we restrict ourselves to one of the simplestpossibilities that leads to nontrivial solutions. To this end, we choose u thatobeys the differential equation

u′ = A(z)u , A(z) =(βz+ 2ρz

). (3.6)

We will show below that the choice (3.6) relates Kiun to un, un±1 that,in turn, allows us to formulate the conditions of cut off for Hamiltonian inthe form of algebraic equations which its coefficients obey. It follows from(3.6) that u = zβ exp(ρz2). To ensure asymptotic analytic behavior nearz = 0, we demand that β = 0, 1, 2, . . . . Now let us take into account somebasic properties of coherent states (see, e.g. Ch. 7 of Ref. [18]). Our func-tions un(z) must belong to the Bargmann-Fock space. It means that they


should obey the conditions of integrability and analyticity. The conditionof integrability for any two functions f , g from our space∫

dzdz∗f∗ge−zz∗<∞ , (3.7)

entails, for our choice of u, |ρ| < 1/2.Taking into account eq. (3.6), it is straightforward to show that

K+un = C+un+1 , (3.8)

K−un = A−(n)un +B−(n)un−1 + C−un+1 , (3.9)

K0un = A0(n)un + C0un+1 , (3.10)

where

C+ =1

2, C− = 2ρ2 , C0 = ρ ,

A−(n) = (2β + 4n+ 1)ρ , A0(n) =1

4(2β + 4n+ 1) ,

B−(n) =1

2(β + 2n)(β + 2n− 1) .

Using eqs.(3.8)–(3.10), one can present the action of the operator (3.1) inthe form

Hun = D2un+2 +D1(n)un+1 + D0(n)un

+ D1(n)un−1 + D2(n)un−2 , (3.11)

where

D2 =a+0

2C0 + a0−C−C0 + a00C

20 + a−−C−C− +

a++

4(3.12)

D1(n) = a−− [A−(n)C− + C−A−(n+ 1)]

+ a00 [A0(n)C0 + C0A0(n+ 1)] +a+0

2A0(n) (3.13)

+ a0−[A−(n)C0 + C−A0(n+ 1)] + a0C0 + a−C− +a+2,

D0 = a00A20(n) + a0−[A−(n)A0(n) +B−(n)C0]

+ a0A0(n) + a−A−(n) , (3.14)

D1 (n) = a−− [A−(n)B− (n) +B− (n)A−(n− 1)]+ a0−B−(n)A0(n− 1) + a−B−(n) , (3.15)

D2 = a−−B− (n)B− (n− 1) . (3.16)

For the operator (3.1) to be quasi-exactly solvable with the invariant sub-space (3.5), it is necessary that the following conditions of cut off besatisfied:

D2 = D1(N) = D1(0) = D2(0) = D2(1) = 0 . (3.17)


4. Explicit examples of invariant subspaces

The analysis of the system (3.17) is rather cumbersome. In this section welist shortly the particular examples of results, when the systems turn outto be QES.

1. u = cosωz, v = sinωz:

H = a00K20 + a0−K0K−

+[a0−2ω2 − 2a00(N + 1)

]K0 + a−K− +

a002ω2K+ (4.1)

2. We also found the following QES case: u = Jn(z), v(z) = Jn+1(z)(Bessel functions):

a− =(3 + 2n)

4a0− , a+0 = 0 = a++ , a+ =

a002,

a0 =a0−2− a00

2(4N + 3) . (4.2)

The effective Hamiltonian is non-Hermitian.3.

a++ = 0, a− =αγ − ρ(4N + 3)

4ρa0−, a00 = f (ρ) a0−,

a0 =(ρ− αγ + N + 1

ρ

)a0− , f(ρ) = −1 + 4ρ

2

2ρ. (4.3)

Our subspace is

span{Ψnz2n exp[(ρ

2

)z2], Ψn+1z

2n+1 exp[

(ρ

2

)z2]} ,

where Ψn is the wave function of the n-th level of the harmonic oscil-lator, Ψn = exp(−1

2ρz2)Hn(z

√ρ), Hn is the Hermite polynomial. Here

β = 0, 1, . . . .

We would like to stress that the corresponding system represents ananharmonic (not harmonic!) Bose oscillator. The functions, which have thesame form as those of an harmonic oscillator, appear in this context in thecoherent state representation (not in the coordinate one, as would be thecase for the usual harmonic oscillator) and represent auxiliary quantities.


5. Concluding remarks

During recent years, the class of QES was extended considerably to includetwo- and many-dimensional systems, matrix models [23], the QES anhar-monic oscillator with complex potentials ([13], p. 192; [26]), etc. Meanwhile,it turned out that, apart from these (sometimes rather sophisticated andexotic) situations, quasi-exact solvability exists in an everyday life aroundus where anharmonic Bose oscillators can be met at every step. In par-ticular, the results obtained can be exploited in solid state or molecularphysics, theory of magnetism, etc. The approach suggested in the presentpaper, shows the line along which a lot of second-quantized models withalgebraization of the part of the spectrum can be constructed. This ap-proach can be also extended to systems with interaction of subsystems ofdifferent nature—in particular, between spin and Bose operator, Bose andFermi oscillators.

References

1. Razavy M 1980 Am. J. Phys. 48 2852. Razavy M 1980 1981 Phys. Lett. A 82 73. Zaslavskii O B, Ulyanov V V and Tsukernik V M 1983 Fiz. Nizk. Temp. 9 511 [Sov.

J. Low Temp. Phys. 9 259]4. Zaslavskii O B and V V Ulyanov 1984 Zh. Eksp. Teor. Phys. 87 1724 [Sov. Phys.

JETP 60 991]5. Zaslavskii O B and Ulyanov V V 1987 Teor. Mat. Fiz. 71 260 [Theor. Math. Phys.

71 520]6. Turbiner A V and Ushveridze A G 1987 Phys. Lett. A 126 1817. Turbiner A V 1988 Zh. Eksp. Teor. Fiz. 94 338. Turbiner A V 1988 Funct. Anal. 22 929. Turbiner A V 1988 Commun. Math. Phys. 118 46710. Zaslavskii O B 1990 Sov. Phys. J. 33 1311. Turbiner A V 1994 Contemp. Math. 160 26312. Ushveridze A G 1994 Quasi-exactly solvable models in quamtum mechanics (Bristol:

Institute of Physics)13. Ulyanov V V and Zaslavskii O B 1992 Phys. Reports 216 17914. Zaslavskii O B 1990 Phys. Lett. A 149 36515. Alvarez G and Alvarez-Estrada R F 1995 J. Phys. A: Math. Gen. 28 576716. Debergh N 2000 J. Phys. A: Math. Gen. 33 710917. Dolya S N and Zaslavskii O B 2000 J. of Phys. A: Math. Gen. 33 L36918. Klauder J R and Sudarshan E C G 1968 Fundamentals of quantum optics (Benjamin

W A, Inc. New York Amsterdam)19. Erdelyi A (ed) 1953 Higher transcendental functions (Bateman Manuscript Project

vol. 1) (New York: Mc-Graw-Hill)20. Brihaye Y and Kosinski P 1994 J. Math. Phys. 35 308921. Zhdanov R 1997 Phys. Lett. B 405 25322. Zhdanov R 1997 J. Phys. A: Math. Gen. 30 876123. Spichak S and Zhdanov R 1999 J. Phys. A: Math. Gen. 32 3815


24. Turbiner A V 1992 J. Math. Phys. 33 398925. Post G and Turbiner A V 1995 Russ. J. Math. Phys. 3 11326. Bender C M and Boettcher S 1998 J. Phys. A: Math. Gen. 31 L273

THE RIEMANN AND EINSTEIN–WEYL GEOMETRIES IN

THE THEORYOF ORDINARYDIFFERENTIAL EQUATIONS,

THEIR APPLICATIONS AND ALL THAT

VALERII DRYUMA ([email protected])∗Institute of Mathematics and Informatics, AS RM, 5 AcademieiStreet,2028 Kishinev, Moldova

Abstract. Some properties of the 4-dimensional Riemannian spaces with metrics

ds2 = 2(za3 − ta4)dx2 + 4(za2 − ta3)dxdy + 2(za1 − ta2)dy

2 + 2dxdz + 2dydt

associated with the second order nonlinear differential equations

y′′ + a1(x, y)y′3 + 3a2(x, y)y

′2 + 3a3(x, y)y′ + a4(x, y) = 0

with arbitrary coefficients ai(x, y) are considered. Three-dimensional Einstein–Weyl spacesconnected with dual equations

b′′ = g(a, b, b′)

where the function g(a, b, b′) satisfies the partial differential equation

gaacc + 2cgabcc + 2ggaccc + c2gbbcc + 2cggbccc + g2gcccc + (ga + cgb)gccc − 4gabc− 4cgbbc − cgcgbcc − 3ggbcc − gcgacc + 4gcgbc − 3gbgcc + 6gbb = 0

are also investigated.The theory of invariants for second order ODE’s is applied to the study of nonlinear

dynamical systems dependent on a set of parameters.

1. Introduction

Second order ODE’s of the type

y′′ + a1(x, y)y′3+ 3a2(x, y)y

′2 + 3a3(x, y)y′ + a4(x, y) = 0 (1.1)

are connected with nonlinear dynamical systems of the form

dx

dt= P (x, y, z, αi),

dy

dt= Q(x, y, z, αi),

dz

dt= R(x, y, z, αi),

∗ Work supported in part by MURST, Italy

115


116 V. DRYUMA

where αi are parameters.For example, the Lorenz system

X = σ(Y −X), Y = rX − Y − ZX, Z = XY − bZ,

which exhibits chaotic behaviour at some values of the parameters, is equiv-alent to the equation

y′′ − 3

yy′2 + (αy − 1

x)y′ + εxy4 − βx3y4 − βx2y3 − γy3 + δy

2

x= 0, (1.2)

where

α = b+σ+1σ , β =

1

σ2, γ =

b(σ + 1)

σ2,

δ = (σ+1)σ , ε =

b(r − 1)σ2

.

The theory of invariants was first used in [1]–[5] for investigating the prop-erties of this system.

Another example is the third-order differential equation

d3X

dt3+ a

d2X

dt2−(dX

dt

)2

+X = 0, (1.3)

which manifests chaotic properties provided the parameter a satisfies 2.017 <a < 2.082 [25]. This equation can be rewritten in the form (1.1)

y′′ +1

yy′2 +

a

yy′ +

x

y2− 1 = 0

with the help of the standard substitution

dX

dt= y(x),

d2X

dt2= y′y,

d3X

dt3= y′′y2 + y′2y.

According to the Liouville theory [6]–[10] all equations of the type (1.1)belong to two different classes

I. ν5 = 0,II. ν5 �= 0.

Here the value ν5 is given by

ν5 = L2(L1L2x − L2L1x) + L1(L2L1y − L1L2y)

− a1L31 + 3a2L

21L2 − 3a3L1L

22 + a4L

32

and L1, L2 are defined by

L1 =∂

∂y(a4y + 3a2a4)−

∂

∂x(2a3y − a2x + a1a4)− 3a3(2a3y − a2x)− a4a1x,

RIEMANN–EINSTEIN–WEYL GEOMETRIES AND ODES 117

L2 =∂

∂x(a1x − 3a1a3) +

∂

∂y(a3y − 2a2x + a1a4)− 3a2(a3y − 2a2x) + a1a4y.

For the equations of the class ν5 = 0, R. Liouville discovered the seriesof semi-invariants

wm+2 = L1∂wm∂y− L2

∂wm∂x

+mwm(∂L2

∂x− ∂L1

∂y

),

starting from :

w1 =1

L41

[L31(α

′L1 − α′′L2) +R1(L21)x − L2

1R1x + L1R1(a3L1 − a4L2)],

w2 =1

L42

[L32(α

′L2 − αL1)−R2(L22)y + L

22R2y − L2R2(a1L1 − a2L2)

].

Here

R1 = L1L2x − L2L1x + a2L21 − 2a3L1L2 + a4L

22

R2 = L1L2y − L2L1y + a1L21 − 2a2L1L2 + a3L

22

α = a2y − a1x + 2(a1a3 − a22)α′ = a3y − a2x + a1a4 − a2a3α′′ = a4y − a3x + 2(a2a4 − a23).

When w1 = 0 there are another series of semi-invariants

i2m+2 = L1∂i2m∂y− L2

∂i2m∂x

+ 2mi2m(∂L2

∂x− ∂L1

∂y

),

where

i2 =3R1

L1+∂L2

∂x− ∂L1

∂y, (1.4)

and a corresponding sequence for absolute invariants

j2m =i2mim2.

For ν5 �= 0 the semi-invariants have the form

νm+5 = L1∂νm∂y− L2

∂νm∂x

+mνm(∂L2

∂x− ∂L1

∂y

),

and the corresponding series of absolute invariants satisfies

[5tm − (m− 2)t7tm−2]ν2/55 = 5(L1∂tm−2∂y

− L2∂tm−2∂x

)(1.5)

118 V. DRYUMA

wheretm = νmν

−m/55 .

From the equations (1.5) it follows that some relations constraining theinvariants are useful for the theory of second order ODE.

In fact, let us consider a relation of the form

t9 = f(t7).

Then we have

(5f(t7)− 7t27)ν2/55 = 5

(L1∂t7∂y− L2

∂t7∂x

),

(5(t11 − 9t7f(t7))ν2/55 = 5f ′(t7)(L1∂t7∂y− L2

∂t7∂x

),

(5t13 − 11t7t11)ν2/55 = 5(L1∂t11∂y− L2

∂t11∂x

),

and(5f − 7t27)f ′t7 = 5t11 − 9f(t7)t7,

from which we get t11 = g(t7).In the simplest case t9 = at

27 we have

t11 = a(2a− 1)t37, t13 = a(2a− 1)(3a − 2)t47,t15 = a(2a− 1)(3a− 2)(4a − 3)t57 , . . . .

These relations show that some values of the parameter a

a = 0, 1/2, 2/3, 3/4, 4/5 , · · ·

are special for the corresponding second order ODE’s.The first application of the Liouville theory for the study of properties

of nonlinear dynamical systems like Lorenz system was presented in theworks of the present author [1]–[5].

In particular for the second order differential equation (1.2) (equivalentto the Lorenz system) the ν5-invariant has the form

ν5 = Ax2 +

B

x2y2+ C

where

A = αβ(10α − α2 − 6δ), B = α(49α

2 + 23αδ − 2δ2

),

C = α(29α

4 + 6εδ − 4αε − α2γ).


In this case the conditionν5 = 0

corresponds toA = 0, B = 0, C = 0.

In particular it leads to the values

σ = −1/5, b = −16/5, r = −7/5,

which have not been considered before in the theory of the Lorenz system.The consideration of the invariants νm+2 is connected with unwieldy

calculations and does not allow us to apply it for the investigation of thissystem. Here we show that is possible to get a more detailed informationfor the case of the equation (1.3).

With this aim we transform the equation

y′′ +1

yy′2 +

a

yy′ +

x

y2− 1 = 0

to a more convenient form. Firstly, the variable x can be expressed as

x = y2y′′ − yy′2 − ayy′ + y2,

and after differentiating this expression we get a third-order ODE whichcan be reduced to the second order ODE

y′′ +1

yy′2 +

(4x+

4

xy

)y′ +

a

x2− 2

xy+

1

x2y2+y

x2= 0.

This equation admits the invariants

L1 =3y + 2a

3x2y2, L2 =

a

xy3,

and

ν5 = −19a3(2a2y + 18xy − 9)/(x5y4),

ν7 =1

27a4(54xy2 − 27y − 20a3y − 180axy + 72a)/(x7y15),

ν9 =2

81a6(702xy2 − 297y − 140a3y − 1260axy + 432a)/(x9y19),

ν11 =4

27a8(990xy2 − 369y − 140a3y + 1260axy + 384a)/(x11y23),

ν13 =40

81a10(2754xy2 − 927y − 308a3y − 2772axy + 768a)/(x13y27),

ν15 =80

243a12(42714xy2 − 13203y − 4004a3y − 36036axy

120 V. DRYUMA

+ 9216a)/(x15y31),

ν25 =985600

6561a22(48428550xy2 − 11175165y − 2704156a3y

− 24337404axy + 4718592a)/(x25y51).From these expressions we can see that only numerical values of coefficientsin the formulas for the invariants are changed under the transition from νmto νm+2.

This fact is useful for studying the relations between the invariantswhen the parameter a is changed. Notice that the starting equation (1.3)is connected with the Painleve I equation for the case a = 0. The firstapplications of the Liouville invariants in the theory of Painleve equationswas presented in the works of author [1]–[6]. In particular, for the equationsof the Painleve type it follows that ν5 = 0. Recently the relations betweenthe invariants for the P-type equations were studied in [30].

2. The Riemann spaces in the theory of ODE’s

Here we present the construction of the Riemann spaces connected withthe equations of type (1.1).

We start from the equations of geodesical lines of two-dimensional spaceA2 equipped with a affine (or Riemann) connection. They have the form

x+ Γ111x2 + 2Γ112xy + Γ

122y

2 = 0,

y + Γ211x2 + 2Γ212xy + Γ

222y

2 = 0.

This system of equations is equivalent to the equation

y′′ − Γ122y′3+ (Γ222 − 2Γ112)y′

2+ (2Γ212 − Γ111)y′ + Γ211 = 0,

which is of type (1.1) for a particular choice of the coefficients ai(x, y).The following proposition holds

Proposition 2.1 The equation (1.1) with the coefficients ai(x, y) describesthe geodesics on a surface with a metric determined by

ds2 =1

Δ2[ψ1dx

2 + 2ψ2dxdy + ψ3dy2],

where Δ = ψ1ψ3 − ψ22, provided the relations

ψ1x + 2a3ψ1 − 2a4ψ2 = 0,

ψ3y + 2a1ψ2 − 2a2ψ3 = 0,

ψ1y + 2ψ2x − 2a3ψ2 + 4a2ψ1 − 2a4ψ3 = 0,

ψ3x + 2ψ2y + 2a2ψ2 − 4a3ψ3 + 2a1ψ1 = 0,


between the coefficients ai(x, y) and the components of the metrics ψi(x, y)are fulfilled.

The equations (1.1) with arbitrary coefficients ai(x, y) may be consid-ered as the equations describing the geodesics of a 2-dimensional spaceA2

x− a3x2 − 2a2xy − a1y2 = 0,

y + a4x2 + 2a3xy + a2y

2 = 0

equipped with a projective connection with components

Π1 =

( −a3 −a2a4 a3

), Π2 =

( −a2 −a1a3 a2

).

The curvature tensor of this type of connection is

R12 =∂Π2

∂x− ∂Π1

∂y+ [Π1,Π2]

and it has the components

R1112 = a3y − a2x + a1a4 − a2a3 = α′,

R1212 = a2y − a1x + 2(a1a3 − a22) = α,

R2112 = a3x − a4y + 2(a23 − a2a4) = −α′′,

R2212 = a2x − a3y + a3a2 − a1a4 = −α′.

In order to construct the Riemann space connected with the equationof type (1.1) we use the Riemannian extension W 4 of the space A2 withconnection Πkij [12] . The corresponding metric is

ds2 = −2Πkijξkdxidxj + 2dξidxi.

In our case it takes the following form (ξ1 = z, ξ2 = t)

ds2 = 2(za3 − ta4)dx2 + 4(za2 − ta3)dxdy+ 2(za1 − ta2)dy2 + 2dxdz + 2dydt. (2.1)

Thus we may state

Proposition 2.2 For a given equation of type (1.1) there exists a Rie-mannian space with metric (2.1) whose geodesics contain integral curves of(1).

122 V. DRYUMA

The calculation of the geodesics of the space W 4 with the metric (2.1)leads to the system of equations

d2x

ds2− a3

(dx

ds

)2

− 2a2dx

ds

dy

ds− a1

(dy

ds

)2

= 0,

d2y

ds2+ a4

(dx

ds

)2

+ 2a3dx

ds

dy

ds+ a2

(dy

ds

)2

= 0,

d2z

ds2+ [z(a4y − α′′)− ta4x]

(dx

ds

)2

+2[za3y − t(a3x + α′′)]dx

ds

dy

ds

+ [z(a2y + α)− t(a2x + 2α′)](dy

ds

)2

+ 2a3dx

ds

dz

ds

− 2a4dx

ds

dt

ds+ 2a2

dy

ds

dz

ds− 2a3

dy

ds

dt

ds= 0,

d2t

ds2+ [z(a3y − 2α′)− t(a3x − α′′)]

(dx

ds

)2

+2[z(a2y − α)− ta2x]dx

ds

dy

ds+ [za1y − t(a1x + α)]

(dy

ds

)2

+2a2dx

ds

dz

ds− 2a3

dx

ds

dt

ds+ 2a1

dy

ds

dz

ds− 2a2

dy

ds

dt

ds= 0,

in which the first two equations for the coordinates x(s), y(s) are equivalentto equation (1.1).

In turn the two last equations for the coordinates z(s) and t(s) have theform of a linear second-order system of differential equations

d2Ψ

ds2+A(x, y)

dΨ

ds+B(x, y)Ψ = 0 (2.2)

where Ψ(x, y) is a two-component vector Ψ1 = z(s), Ψ2 = t(s) and A(x, y)and B(x, y) are 2× 2 matrix-functions.

Notice that the complete system of equations has the first integral

2(za3 − ta4)x2 + 4(za2 − ta3)xy + 2(za1 − ta2)y2 + 2xz + 2yt = 1,

which is equivalent to the relation

zx+ ty =s

2+ μ,

where μ is an integration constant. This allows us to use only one linearsecond-order differential equation from the matrix system (2.2) in the studyof concrete examples.


Thus, we have constructed the four-dimensional Riemannian space withthe metric (2.1) and with the connection

Γ1 =

⎛⎜⎜⎝−a3 −a2 0 0a4 a3 0 0

z(a4y − α′′)− ta4x za3y − t(a3x + α′′) a3 −a4z(a3y − 2α′)− t(a3x − α′′) z(a2y − α)− ta2x a2 −a3

⎞⎟⎟⎠ ,

Γ2 =

⎛⎜⎜⎝−a2 −a1 0 0a3 a2 0 0

za3y − t(a3x + α′′) z(a2y + α) − t(a2x + 2α′) a2 −a3z(a2y − α)− ta2x za1y − t(a1x + α) a1 −a2

⎞⎟⎟⎠ ,

Γ3 =

⎛⎜⎜⎝0 0 0 00 0 0 0a3 a2 0 0a2 a1 0 0

⎞⎟⎟⎠ , Γ4 =

⎛⎜⎜⎝0 0 0 00 0 0 0−a4 −a3 0 0−a3 −a2 0 0

⎞⎟⎟⎠ .The curvature tensor of this metric is

R1112 = −R3

312 = −R2212 = R

4412 = α

′, R1212 = −R4

312 = α,

R2112 = −R3

412 = −α′′, R1312 = R

1412 = R

2312 = R

2412 = 0,

R3112 = 2z(a2α

′′ − a3α′) + 2t(a4α′ − a3α′′),R4212 = 2z(a3α

′ − a2α) + 2t(a3α− a2α′),R3212 = z(αx − α′y + a1α′′ − a3α) + t(α′′y − α′x + a4α− a2α′′),

R4112 = z(α′y − αx + a1α′′ − a3α) + t(α′x − α′′y + a4α− a2α′′).

By using the expressions for the components of the projective curvature ofthe space A2

L1 = α′′y − α′x + a2α′′ + a4α− 2a3α′,L2 = α′y − αx + a1α′′ + a3α− 2a2α′,

they can be rewritten as

R4112 = z(L2 + 2a2α

′ − 2a3α)− t(L1 + 2a3α′ − 2a4α),

R3212 = z(−L2 + 2a1α

′′ − 2a2α′) + t(L1 + 2a3α′ − 2a2α′′),

Rij34 = 0,

R13 =

⎛⎜⎜⎝0 0 0 00 0 0 00 −α′ 0 0α′ 0 0 0

⎞⎟⎟⎠ , R14 =

⎛⎜⎜⎝0 0 0 00 0 0 00 α′′ 0 0−α′′ 0 0 0

⎞⎟⎟⎠ ,

124 V. DRYUMA

R23 =

⎛⎜⎜⎝0 0 0 00 0 0 00 −α 0 0α 0 0 0

⎞⎟⎟⎠ , R24 =

⎛⎜⎜⎝0 0 0 00 0 0 00 α′ 0 0−α′ 0 0 0

⎞⎟⎟⎠ .The Ricci tensor Rik = R

lilk of our space D

4 has the components

R11 = 2α′′, R12 = 2α′, R22 = 2α,

and the scalar curvature R = gingkmRnm of the space D4 is R = 0.Now we can introduce the tensor

Lijk = ∇kRij −∇jRik = Rij;k −Rik;j.

It has the following components

L112 = −L121 = 2L1, L221 = L212 = −2L2

which allow us to construct the invariant conditions connected with theequations (1.1) from the covariant derivatives of the curvature tensor andthe values of L1, L2.

The Weyl tensor of the space D4 is

Clijk = Rlijk +1

2(gjlRik + gikRjl − gjkRil − gilRjk) +

R

6(gjkgil − gjlgik).

It has only one component

C1212 = tL1 − zL2.

Observe that the values of L1 and L2 in this formulas are the same as theLiouville expressions in theory of invariants of the equations (1.1).

Using the components of the Riemann tensor

R1412 = α′′, R2412 = α

′, R2312 = −α, R3112 = α′,

R1212 = z(αx−α′y+a1α′′− 2a2α′+a3α)+ t(α′′y −α′x+a4α− 2a3α′−a2α′′)the equation ∣∣ RAB − λgAB ∣∣ = 0

for determining the Petrov type of the spaces D4 has been considered. HereRAB is a symmetric 6× 6 matrix constructed from the components of theRiemann tensor Rijkl of the space D

4.In particular we have checked that all the scalar invariants of the space

D4 of the formRijR

ij = 0, RijklRijkl = 0, . . .


constructed from the curvature tensor of the space M4 and its covariantderivatives vanish.

Remark. The spaces with metrics (2.1) are flat for the equations (1.1)under the conditions

α = 0, α′ = 0, α′′ = 0,

for the coefficients ai(x, y).Such type of equations imply

L1 = 0, L2 = 0

and they can be reduced to the the form y′′ = 0 with the help of appropriatepoint transformations.

On the other hand there are examples of equations (1.1) such that L1 =L2 = 0 and

α �= 0, α′ �= 0, α′′ �= 0.

For such type of equations the curvature of the corresponding Riemannspaces does not vanish.

In fact, the equation

y′′ + 2eϕy′3 − ϕyy′2 + ϕxy′ − 2eϕ = 0 (2.3)

verifies L1 = L2 = 0 and

α �= 0, α′ �= 0, α′′ �= 0.

provided the function ϕ(x, y) is a solution of the Wilczynski-Tzitzeikanonlinear equation integrable by the Inverse Transform Method

ϕxy = 4e2ϕ − e−ϕ. (2.4)

In particular, there are non-flat Riemannian spaces even for linear second-order differential equations.

Remark. The study of the properties of Riemann spaces with the metric(2.1) associated to equations (1.2) possesing chaotical behaviour at thevalues of coefficients (σ = 10, b = 8/3, r > 24) is an important problem.The spaces with such values of the parameters verify special relations forthe components of the curvature tensor.

In order to study this problem we may use the geodesic deviationequation

d2ηi

ds2+ 2Γilm

dxm

ds

dηl

ds+∂Γikl∂xj

dxk

ds

dxl

dsηj = 0

126 V. DRYUMA

where Γilm are the Christoffell coefficients of the metric (2.1) with thecoefficients

a1 = 0, a2 = −1

y, a3 =

(αy3− 1

3x

),

a4 = εxy4 − βx3y4 − βx2y3 − γy3 + δy

2

x.

For the equations

y′′ + a4(x, y) = 0

the four-dimensional Riemann spaces with the metric

ds2 = −2ta4dx2 + 2dxdz + 2dydt

and geodesic equations

x = 0, y + a4(x, y)(x)2 = 0, t+ a4y(x)

2t = 0

z − ta4x(x)2 − 2ta4y xy − 2a4xt = 0

are connected.It is interesting to note that for the Painleve II equation

y′′ = 2y3 + xy + α

the system for geodesic deviations of the corresponding Riemann space

d2η1

ds2= 0,

d2η2

ds2= (6y2 + s)η2 + 4y3 + 3sy + α,

d2η3

ds2= −(12ty2 + 2ts)dη

ds− (4y3 + 2sy + 2α)dη

4

ds

−(t+ 24tyy + 2y2t+ 2s)η2 − (y + 12y2y + 2sy)η4−2ty − 4tsy − 12ts2y − 4y3t− 4syt− 2αt,

d2η4

ds2= (6y2 + s)η4 + 12ty2 + 3ts+ 12tyη2

depends on the parameter α.For the equations

y′′ + 3a3(x, y)y′ + a4(x, y) = 0

the corresponding Riemann spaces have the metrics

ds2 = 2(za3 − ta4)dx2 − 4ta3dxdy + 2dxdz + 2dydt


and the geodesic equations

x− a3x2 = 0,

y + 2a3xy + a4x2 = 0,

t− 2a3xt− a3yx2z + (a4y − 2a32 − 2a3x)x2t = 0,

z + 2a3xz − 2(a3y + a4x)t+ [(a3x + 2a32)x2 + 2a3yxy]z−[a4xx2 + 2(a4y − 2a23)xy + 2a3y y2]t = 0.

Let us consider the possibility of submersion of the spaces with metrics(2.1) from the point of view of the theory of the embedding of Riemannspaces into flat spaces

For the Riemann spaces of the first class (which can be embedded intothe 5-dimensional Euclidean space) the following conditions are fulfilled

Rijkl = bikbjl − bilbjk, bij;k − bik;j = 0

where Rijkl are the components of the curvature tensor associated with themetric ds2 = gijdx

idxj .The consideration of these relations for the spaces with the metric (2.1)

leads to conditions on the values ai(x, y). Thus one deduces that the em-bedding in the 5-dimensional flat space with the flat metrics is possible onlyin the case

ai(x, y) = 0.

For the spaces of the second class (which admit the embedding intothe 6-dimensional flat space with appropriate signature) the conditions aremore complicated. They are

Rabcd = e1(ωacωbd − ωadωbc) + e2(λacλbd − λadλbc),ωab;c − ωac;b = e2(τcλab − τbλac),λab;c − λac;b = −e1(τcωab− τbωac),τa;b − τa;c = ωacλ

cb − λacωcb .

and lead to the relations

εabcdεnmrsεpqikRabnmRcdpqRrsik = 0,

εcdmnRabcdRabmn = −8e1e2εcdmnτc;dτm;n.

3. On the relationship with the theory of surfaces

The existence of the Riemann metrics for the equations (1.1) may be usefulfor constructing the corresponding surfaces.

128 V. DRYUMA

A possible application is the study of two-dimensional subspaces of agiven 4-dimensional space which generalize the surfaces of translation. Theequations for coordinates Zi(u, v) of such a type of surfaces are

∂2Zi

∂u∂v+ Γijk

∂Zj

∂u

∂Zk

∂v= 0. (3.1)

where Γjki are the connection components.Let us consider the system (3.1) in detail. We get the following system

of equations for the coordinates x = x(u, v), y = y(u, v), z = z(u, v),t = t(u, v)

xuv − a3xuxv − a2(xuyv + xvyu)− a1yuyv = 0

yuv + a4xuxv + a3(xuyv + xvyu) + a2yuyv = 0

zuv + zu(a2yv + a3xv) + zv(a2yu + a3xu)− tu(a3yv + a4xv)− tv(a4xu + a3yu) + z(2yuyva1a3 − yvyua1x + xvxua3x − 2xuxva2a4

+ yuxva3y − yvxua3y − 2yvyu(a2)2 + 2yvyua2y + 2xvyu(a3)2)+ t(yvyua2x − xvxua4x − xvyua4y + 2yuyva2a3 − 2yvxua2a4

− 2yvyua3y + 2xvyu(a3)2 + 2yvxu(a3)2 − yvxua4y − 2yuxva2a4− 2yvyua1a4) = 0

tuv + zu(a2xv + a1yv) + zv(a2xu + a1yu)− tu(a3xv + a2yv)− tv(a3xu + a2yu) + z(− 2xuyva1a3 + xvyua1x + 2xvxua2x

+ 2xuxva2a3 + yvyua1y + 2yvxu(a2)2 − 2xvxua4a1 − xvxua3y

+ 2xvyu(a2)2 − 2xvyua1a3 + yvxua1x) + t(− yvxua2x − xvyua2x

+ xvxua4y + 2xuxva2a4 − 2yuyva1a3 − 2xvxua3x − 2xvxu(a3)2+ 2yvyu(a2)

2 − yvyua2y) = 0

Here the following expressions for the Christoffell coefficients were used

Γ111 = −a3(x, y), Γ211 = a4(x, y), Γ112 = −a2(x, y), Γ212 = a3(x, y),

Γ313 = a3(x, y),Γ413 = a2(x, y), Γ314 = −a4(x, y), Γ414 = −a3(x, y),

Γ122 = −a1(x, y), Γ222 = −a2(x, y), Γ323 = a2(x, y),

Γ423 = a1(x, y), Γ324 = −a3(x, y), Γ424 = −a2(x, y),

Γ311 = z∂a3(x, y)

∂x− t∂a4(x, y)

∂x+ 2za3(x, y)

2 − 2za2(x, y)a4(x, y),

Γ411 = 2z∂a2(x, y)

∂x− 2t∂a3(x, y)

∂x− z ∂a3(x, y)

∂y+ t∂a4(x, y)

∂y


+ 2za3(x, y)a2(x, y)− 2za1(x, y)a4(x, y)+ 2ta2(x, y)a4(x, y)− 2ta3(x, y)2,

Γ312 = z∂a3(x, y)

∂y− t∂a4(x, y)

∂y+ 2ta3(x, y)

2 − 2ta2(x, y)a4(x, y),

Γ412 = z∂a1(x, y)

∂x− t∂a2(x, y)

∂x+ 2za2(x, y)

2 − 2za1(x, y)a3(x, y),

Γ422 = z∂a1(x, y)

∂y− t∂a2(x, y)

∂y+ 2ta3(x, y)

2 − 2ta1(x, y)a3(x, y),

Γ322 = 2z∂a2(x, y)

∂y− 2t∂a3(x, y)

∂y− z ∂a1(x, y)

∂x+ t∂a2(x, y)

∂x

+ 2za3(x, y)a1(x, y)− 2ta1(x, y)a4(x, y)+ 2ta2(x, y)a3(x, y)− 2za2(x, y)2.

We notice that the two last equations are linear and reduce to the 2 × 2matrix Laplace equations

∂2Ψ

∂u∂v+A

∂Ψ

∂u+B

∂Ψ

∂v+ CΨ = 0, (3.2)

which can be integrated by means of a generalization of the Laplace trans-formation [26].

For this aim we use the transformations

Ψ1 = (∂v +A)Ψ, (∂u +B)Ψ1 = hΨ,

where the Laplace invariants are

H = Au +BA− C, K = Bv +AB −C

and then construct a new equation of type (2.3) for the function Ψ1 withnew invariants given by

A1 = HAH−1 −HvH−1, B1 = B,

C1 = Bv −H + (HAH−1 −HvH−1)B,

Let us consider some examples. For the first example with set

x = x, y = y, u = x, v = y,

z = z(u, v) = z(x, y), t = t(u, v) = t(x, y).

From the first equations of the full system we get

a2(x, y) = 0, a3(x, y) = 0,

130 V. DRYUMA

and from the next two we have the system of equations

∂2z

∂x∂y− ∂a4(x, y)

∂yt− a4(x, y)

∂t

∂y= 0,

∂2t

∂x∂y+∂a1(x, y)

∂xz + a1(x, y)

∂z

∂x= 0,

They are equivalent to the independent relations

∂z

∂x− ta4(x, y) = 0,

∂t

∂y+ za1(x, y) = 0,

or

∂2z

∂x∂y− 1

a4

∂a4(x, y)

∂y

∂z

∂x+ a1(x, y)a4(x, y)z = 0,

∂2t

∂x∂y− 1

a1

∂a1(x, y)

∂x

∂t

∂y+ a1(x, y)a4(x, y)t = 0.

Any solution of this system of equations supplies examples of surfacescorresponding to ODE’s of the form

d2y

dx2+ a1(x, y)

(dy

dx

)3

+ a4(x, y) = 0

The next example assumes the conditions:

x = u+ v, y = uv.

From the first relation we get

a3 + xa2 = −ya1, ya2 + xa3 = −1− a4

from which it follows

a2 =1 + a4(x, y)− xya1(x, y)

x2 − y , a3 =y2a1(x, y)− x− xa4(x, y)

x2 − y .

As a result we get the equations

d2y

dx2+ a1

(dy

dx

)3

+ 3(1 + a4 − xya1)

x2 − y

(dy

dx

)2

+3(y2a1 − x− xa4)

x2 − ydy

dx+ a4 = 0


In the particular case

a1(x, y) = 0, a4(x, y) =−x2y

we get the equation

d2y

dx2− 3

y

(dy

dx

)2

+3x

y

dy

dx− x

2

y= 0 (3.3)

and the following equations for the coordinates of the corresponding sur-faces

tuv −tuu− tvv− 1

uv(zu + zv) +

u+ v

u2v2z +

uv + u2 + v2

u2v2t = 0,

zuv +zuu+zvv+ (u+ v)

(tvv+tuu

)− uv + u

2 + v2

u2v2z

− u2v + u3 + v3 + v2u

u2v2t = 0.

Observe that the equation (3.3) can be rewritten as

d2z

dρ2− 3

z

(dz

dρ

)2

+

(3

z− 9

)dz

dρ− 10z + 6− 1

z= 0 (3.4)

with the help of the substitution

y(x) = x2z(log(x)).

Another possible application for the study of two-dimensional surfacesin the space with metrics (2.1) is connected with the choice

x = x, y = y, z = z(x, y), t = t(x, y).

Using the expressions

dz = zxdx+ zydy, dt = txdx+ tydy

we get the metric

ds2 = 2(zx + za3 − ta4)dx2 + 2(tx + zy + 2za2 − 2ta3)dxdy+2(ty + za1 − ta2)dy2.

We can use this representation for the analysis of particular cases of equa-tions (1.1).

132 V. DRYUMA

The constraints

zx + za3 − ta4 = 0,

tx + zy + 2za2 − 2ta3 = 0

ty + za1 − ta2 = 0

are connected with flat surfaces and reduce under the substitution

z = Φx, t = Φy

to the system

Φxx = a4Φy − a3Φx,Φxy = a3Φy − a2Φx,Φyy = a2Φy − a1Φx,

which is compatible with the conditions

α = 0, α′ = 0, α′′ = 0.

Remark. The choice of the functions z = Φx, t = Φy satisfying to thesystem of equations

Φxx = a4Φy − a3Φx,Φyy = a2Φy − a1Φx

corresponds to coefficients ai(x, y) which are given by

a4 = Rxxx, a3 = −Rxyy, a2 = Rxyy, a1 = Ryyy

where the function R(x, y) is the solution of WDVV-equation

RxxxRyyy −RxxyRxyy = 1,

and are associated to the equation (1.1)

y′′ −Ryyyy′3 + 3Rxyyy′2 − 3Rxxyy′ +Rxxx = 0.

The following choice of the coefficients ai

a4 = −2ω, a1 = 2ω, a3 =ωxω, a2 = −

ωyω

leads to the system

Φxx +ωxωΦx + 2ωΦy = 0,


Φyy + 2ωΦx +ωyωΦy = 0

the compatibility condition of which

∂2 logω

∂x∂y= 4ω2 +

κ

ω

is the Wilczynski-Tzitzeika-equation.

Remark. The linear system of equations for the WDVV-equation de-termines some surfaces in three-dimensional projective space. In canonicalform it becomes [13]

Φxx −RxxxΦy +(Rxxxy

2−R2xxy

4− RxxxRxxy

2

)Φ = 0,

Φyy −RyyyΦx +(Ryyyx

2−R2xyy

4− RyyyRxxy

2

)Φ = 0.

The relations between Wilczynsky invariants for the linear system cor-responds to several types of surfaces. Some of them are connected withsolutions of the WDVV equation.

Remark. From an elementary point of view the surfaces connected witha system of equations like the Lorenz system can be constructed from theabove scheme.

Under the assumption

z = z(x, y)

we get

σ(y − x)zx + (rx− y − zx)zy = xy − bz.

The solutions of this equation give us examples of surfaces z = z(x, y).The Riemann metric of the space associated with the equation 2 has

the form

1

2ds2 =

(α3zy − z

3x− δy2 t

x− εtxy4 + βtx3y4 + βtx2y3 + γty3

)dx2

+ 2(− zy− α3ty +

t

3x

)dxdy +

t

ydy2 + dxdz + dydt.

The properties of a space with such a metric depend on the parameters α,β, γ δ, ε and may be very specifical when the Lorenz dynamical system hasa strange attractor.

134 V. DRYUMA

4. Symmetry, the Laplace-Beltrami equation, tetradic presentation

Let us consider the system of equations

ξi,j + ξj,i = 2Γkijξk

for the Killing vectors of the metric (2.1). It has the form

ξ1x = −a3ξ1 + a4ξ2 + (zA− ta4x)ξ3 + (zE + tF )ξ4,ξ2y = −a1ξ1 + a2ξ2 + (zC + tD)ξ3 + (za1y − tH)ξ4,

ξ1y + ξ2x = 2[−a2ξ1 + a3ξ2 + (za3y − tB)ξ3 + (zG − ta2x)ξ4,ξ1z + ξ3x = 2[a3ξ3 + a2ξ4], ξ1t + ξ4x = 2[−a4ξ3 − a3ξ4],ξ2z + ξ3y = 2[a2ξ3 + a1ξ4], ξ2t + ξ4y = −2[a3ξ3 − a2ξ4],

ξ3z = ξ4t = 0.

In the particular case

ξ3 = ξ4 = 0, ξi = ξi(x, y)

we get the system of equations

ξ1x = −a3ξ1 + a4ξ2, ξ2y = −a1ξ1 + a2ξ2,

ξ1y + ξ2x = 2[−a2ξ1 + a3ξ2]which is equivalent to the system for the z = z(x, y) and t = t(x, y)

Remark. The Laplace-Beltrami operator

Δ = gij( ∂2

∂xi∂xj− Γkij

∂

∂xk

)can be used for investigating the properties of the metric (2.1). For examplethe equation

ΔΨ = 0

has the form

(ta4 − za3)Ψzz + 2(ta3 − za2)Ψzt + (ta2 − za1)Ψtt +Ψxz +Ψyt = 0.

Some solutions of this equation are connected with the geometry of themetric (2.1).

By substitutingΨ = exp(zA+ tB)

into the equationΔΨ = 0,


we get the conditionsA = Φy, B = −Φx,

and

a4Φ2y − 2a3ΦxΦy + a2Φ2

x − ΦyΦxx +ΦxΦxy = 0,

a3Φ2y − 2a2ΦxΦy + a1Φ2

x −ΦyΦxy +ΦxΦyy = 0.

Another possible application for the study of the properties of a givenRiemann space is connected with the computation of the heat invariantsof the Laplace-Beltrami operator. For that aim the fundamental solutionK(τ, x, y) of the heat equation

∂Ψ

∂τ= gij

(∂2Ψ

∂xi∂xj− Γkij

∂Ψ

∂xk

)

is considered. The diagonal of the function K(τ, x, y) has the followingasymptotic expansion as t→ 0+

K(τ, x, x) ∼∞∑n=0

an(x)τn−2

where the coefficients an(x) are local invariants (heat invariants) of theRiemann space D4 with the metric (2.1).

In turn the eikonal equation

gij∂F

∂xi∂F

∂xj= 0

or

FxFz + FyFt − (ta4 − za3)FzFz − 2(ta3 − za2)FzFt − (ta2 − za1)FtFt = 0,

can also be used for investigating the properties of isotropic surfaces in thespace with metric (2.1).

In particular, the solutions of the eikonal equation of the form

F = A(x, y)z2 +B(x, y)zt+ C(x, y)t2 +D(x, y)z + E(x, y)t

lead to the following constraints for the coefficients

2AAx +BAy − a1B2 − 4a2AB − 4a3A2 = 0,

2ABx +BAx + 2CAy +BBy − 4a1BC − a2(B2 + 8AC) + 4a4A2 = 0,

2CBy +BCy + 2ACx +BBx − 4a1C2 + a3(B2 + 8AC) + 4a4AB = 0,

2CCy +BCx + 4a2C2 + 4a3BC + a4B

2 = 0,

136 V. DRYUMA

2ADx +DAx + EAy +BDy − 2a1BE − 2a2(BD + 2AE)− 4a3AD = 0,

2CDy+(BD)x+2AEx+(BE)y−4a1EC−4a2CD+4a3AE+4a4AD = 0,

2CEy + CEy +DCx +BEx − 4a2CE + 2a3(BE + 2CD) + 2a4BD = 0,

DDx + EDy − a1E2 − 2a2DE − a3D2 = 0,

EEy +DEx + a2E2 + 2a3DE + a4D

2 = 0

which may be useful for the theory of the equations (1.1).

Remark. The metric (2.1) has a tetradic presentation

gij = ωai ωbjηab

where

ηab =

⎛⎜⎜⎝0 0 1 00 0 0 11 0 0 00 1 0 0

⎞⎟⎟⎠ .For example we get

ds2 = 2ω1ω3 + 2ω2ω4

where

ω1 = dx+ dy, ω2 = dx+ dy +1

t(a2 − a4)(dz − dt),

ω4 = −t(a4dx+ a2dy), ω3 = z(a3dx+ a1dy) +1

(a2 − a4)(a2dz − a4dt).

and

a1 + a3 = 2a2, a2 + a4 = 2a3.

Remark. Some of the equations on the curvature tensors in the spaceM4 are connected with ODE’s. For example, the equation

Rij;k +Rjk;i +Rki;j = 0

leads to the following conditions on the coefficients ai(x, y)

α′′x + 2a3α′′ − 2a4α′ = 0,

αy + 2a1α′ − 2a2α = 0,

α′′y + 2α′x + 4a2α

′′ − 2a4α− 2a3α′ = 0,

αx + 2α′y − 4a3α+ 2a2α′ + 2a1α′′ = 0.


The solutions of this system give us the second-order equations associatedto the space M4 with a given condition on the Ricci tensor. The simplestexamples are

y′′− 3

2yy′2+y3 = 0, y′′− 3

yy′2+y4 = 0, y′′+3(2+y)y′+y3+6y2−16 = 0.

It is interesting to note that the above system is the Liouville system forgeodesics in the Proposition 1.

The study of invariant conditions like

Rij;k −Rjk;i = Rnijk;n, �Rijkl = 0, �Rijkl;m = 0

is also of interest for the theory of the equations (1.1).

Remark. The construction of the Riemannian extension of two-dimen-sional spaces connected with ODE’s of type (1.1) can be generalized to thethree-dimensional case by equations of the form

x+A1(x)2 + 2A2xy + 2A3xz +A4(y)

2 + 2A5yz +A6(z)2 = 0,

y +B1(x)2 + 2B2xy + 2B3xz +B4(y)

2 + 2B5yz +B6(z)2 = 0,

z + C1(x)2 + 2C2xy + 2C3xz + C4(y)

2 + 2C5yz + C6(z)2 = 0,

or

x′′ + a0 + a1x′ + a2y′ + a3x′2+ a4x

′y′ + a5y′2

+ x′(b0 + b1x′ + b2y′ + b3x′2+ b4x

′y′ + b5y′2) = 0,

y′′ + c0 + c1x′ + c2y′ + c3x′2+ c4x

′y′ + c5y′2

+ y′(b0 + b1x′ + b2y′ + b3x′2+ b4x

′y′ + b5y′2) = 0,

where ai, bi, ci are functions of variables x, y, z.The corresponding expression for the 6-dimensional metric is:

1

2ds2 = −(A1u+B1v + C1w)dx

2 − 2(A2u+B2v + C2w)dxdy

− 2(A3u+B3v + C3w)dxdz − (A4u+B4v + C4w)dy2

− 2(A5u+B5v + C5w)dydz − (A6u+B6v + C6w)dz2

+ dxdu+ dydv + dzdw

This gives us the possibility of studying the properties of such type ofequations from a geometrical point of view.

Let us consider some examples.

138 V. DRYUMA

5. The Riemann metrics of zero curvature and the KdV equation

Let us consider the system of matrix equations

∂Γ2

∂x− ∂Γ1

∂y+ [Γ1,Γ2] = 0,

∂Γ3

∂x− ∂Γ1

∂z+ [Γ1,Γ3] = 0, (5.1)

∂Γ3

∂y− ∂Γ2

∂z+ [Γ2,Γ3] = 0,

where Γi(x, y, z) is the 3× 3 matrix function with matrix elements (Γi)kj =Γkij = Γkji.

This system can be regarded as the zero-curvature condition of somethree-dimensional spaces with the affine connection given by the coefficientsΓ(x, y, z).

If Γkij(x, y, z) are of the form

2Γ1 = y2B1(x, z) + yA1(x, z) + C1(x, z) +1

yD1(x, z),

Γ3 = y2B3(x, z) + yA3(x, z) + C3(x, z) +1

yD3(x, z) +

1

y2E3(x, z),

Γ2 = C2(x, z) +1

yD2(x, z),

then after substitution of these expressions in formulas (5.1) we get thesystem of nonlinear equations for the components of the affine connection.Some of these equations may be useful for applications.

Let us consider the space with metric

gik =

⎛⎝ y2 0 y2l(x, z) +m(x, z)0 0 1

y2l +m 1 y2l2 − 2ylx + 2l m+ 2n

⎞⎠ ,where l,m, n are functions of (x, z). By using the corresponding connection

Γkij =1

2gkl(∂igjl + ∂jgil − ∂lgij)

we get the following matrices Γi

Γ1 =

⎛⎜⎝ yl + my

1y yl2 + lm

y

−γ1 −my −γ2−y 0 −ly

⎞⎟⎠ ,


Γ2 =

⎛⎜⎝1y 0 l

y

−my 0 − lmy − lx0 0 0

⎞⎟⎠ ,

Γ3 =

⎛⎜⎝ l2 + my

ly γ3

γ2 −lx − lmy −γ4

−yl 0 −yl2 + lx

⎞⎟⎠ ,where

γ1 = y2lx − 2yn+m2/y −mxγ2 = y2llx − 2yln+ lm2/y + ylxx −mlx − lmx − nxγ3 = lz +mz/y

2 − 2llx + lxx/y − 2mlx/y2 − lmx/y2 − nx/y2 + yl3+ l2m/y

γ4 = y2l2lx + yllxx − 2lmlx − l2mx − lnx +mmz/y2 +mlxx/y− 2m2lx/y

2 − lmmx/y2 −mnx/y2 − 2yl2x − 2ynl2 + 2nlx+m2l2/y + ylzx − nz.

If l(x, z) = n(x, z) we get

R1313 =( ∂3l∂x3− 3l ∂l

∂x+∂l

∂z

)y2 +

( ∂2m∂x∂z

− 2m ∂2l

∂x2− l ∂

2m

∂x2

− 3∂m∂x

∂m

∂x− ∂2l

∂x2

)y −m∂m

∂z+ 2m2 ∂l

∂x+m

∂l

∂x

+ml∂m

∂x−m∂m

∂z+ 2m2 ∂l

∂x+m

∂l

∂x+ml

∂m

∂x,

and

R1323 =1

y

(− ∂m∂z

+ 2m∂l

∂x+ l∂m

∂x+∂l

∂x

)From the condition Rijkl = 0 it follows that the function l(x, z) solves theKdV-equation

∂3l

∂x3− 3l ∂l

∂x+∂l

∂z= 0,

so that all flat metrics of such a type are determined from solutions of thisequation.

Notice that starting from the Riemannian extensions of a space witha given metric the corresponding metric of the six-dimensional space canbe written. The equations of the geodesics of such 6-dimensional spacecontains the linear second order ODE (Schrodinger operator) which appliesfor integrating the KdV equation.

140 V. DRYUMA

6. Applications in the theory of general Relativity

The Riemann extensions of a given metric can be used for the study ofRiemannian spaces satisfying the Einstein conditions

Rij = gklRijkl = 0

on the curvature tensor Rijkl.Let us consider some examples. Let

ds2 = −t2p1dx2 − t2p2dy2 − t2p3dz2 + dt2 (6.1)

be the metric of the Kasner type in the classical theory of gravitation.The Ricci tensor R ≡ (Rij) of this metric is

R =

⎛⎜⎜⎜⎜⎝p1+p2+p3−1t2p1−2

0 0 0

0 p1+p2+p3−1t2p2−2

0 0

0 0 p3(p1+p2+p3−1)t2p3−2

0

0 0 0p1+p2+p3−p21−p22−p23

t2

⎞⎟⎟⎟⎟⎠ ,and in the particular case R = 0 we get the well-known Kasner solution ofthe vacuum Einsten equations.

Now we shall consider the construction of Riemann extension for themetric (6.1). One gets the eight-dimensional space with local coordinates(x, y, z, t, P,Q,R, S) and the metric

ds2 = −2Γkijξkdxidxj + 2dxdP + 2dydQ + 2dzdR + 2dtdS (6.2)

were Γkij are the Christoffel coefficients of the metric (6.1) and ξk = (P,Q,R, S).They are given by:

Γ411 = p1t2p1−1, Γ422 = p2t

2p2−1, Γ433 = p3t2p3−1,

Γ114 = p1/t, Γ224 = p2/t, Γ334 = p3/t.

As a result we find the metric of the space K8 in the form

ds2 = −2p1t2p1−1Sdx2 − 2p2t2p2−1Sdy2 − 2p3t2p3−1Sdz2−4p1/tPdxdt − 4p2/tQdydt− 4p3/tRdzdt + 2dxdP+2dydQ+ 2dzdR + 2dtdS.

The nonzero components of the Ricci tensor 8Rij are

R11 = 2p1t2p1−2(p1 + p2 + p3 − 1), R22 = 2p2t

2p2−2(p2 + p2 + p3 − 1),

R33 = 2p3t2p3−2(p2+ p2+ p3− 1), R44 = 2(p2+ p2+ p3− p21− p22− p23)/t2


which coincide with the components of the Ricci tensor 4Rij of the spaceK4.

Therefore the geometry of the Riemann space before and after extensionis the same.

In turn the equations of the geodesics of the extended space

d2x

ds2+ 2

p1t

dx

ds

dt

ds= 0,

d2y

ds2+ 2

p2t

dy

ds

dt

ds= 0,

d2z

ds2+ 2

p3t

dz

ds

dt

ds= 0

d2t

ds2+ p1t

2p1−1(dx

ds

)2

+ p2t2p2−1

(dy

ds

)2

+ p3t2p3−1

(dz

ds

)2

= 0,

d2R

ds2− 2p3

t

dt

ds

dR

ds− 2p3t2p3−1

dz

ds

dS

ds+

[2p1p3t

2p1−1

t

(dx

ds

)2

+ 2p2p3t

2p2−1

t

(dy

ds

)2

+ 2p23t

2p3−1

t

(dz

ds

)2

+ 2p3t2

(dt

ds

)2 ]R

+ 2p3t

2p3−1

t

dz

ds

dt

dsS = 0,

d2Q

ds2− 2p2

t

dt

ds

dQ

ds− 2p2t2p2−1

dy

ds

dS

ds+

[2p1p2t

2p1−1

t

(dx

ds

)2

+ 2p2p3t

2p3−1

t

(dz

ds

)2

+ 2p22t

2p2−1

t

(dy

ds

)2

+ 2p2t2

(dt

ds

)2 ]Q

+ 2p2t

2p2−1

t

dy

ds

dt

dsS = 0,

d2P

ds2− 2p1

t

dt

ds

dP

ds− 2p1t2p1−1

dx

ds

dS

ds+

[2p1p3t

2p1−1

t

(dz

ds

)2

+ 2p2p1t

2p2−1

t

(dy

ds

)2

+ 2p21t

2p1−1

t

(dx

ds

)2

+ 2p1t2

(dt

ds

)2 ]P

+ 2p1t

2p1−1

t

dx

ds

dt

dsS = 0,

d2S

ds2− 2p3

t

dz

ds

dR

ds− 2p2

t

dy

ds

dQ

ds− 2p1

t

dx

ds

dP

ds+4p22t2dy

ds

dt

dsQ

+4p23t

dz

ds

dt

dsR+

[p1(2p1 − 1)t2p1−1

t

(dx

ds

)2

+p2(2p2 − 1)t2p2−1

t

(dy

ds

)2

+p3(2p3 − 1)t2p3−1

t

(dz

ds

)2 ]S = 0,

contain the linear 4 × 4 matrix system of second-order ODE’s for theadditional coordinates (P,Q,R, S)

d2Ψ

ds2= A(x, y, z, t)

dΨ

ds+B(x, y, z, t)Ψ.

142 V. DRYUMA

Here A,B are 4×4 matrix functions depending on the coordinates (x, y, z, t).This fact allows us to use the methods of soliton theory for integrating thefull system of geodesics and its corresponding Einstein equations.

Observe that the signature of the space 8D is 0, i.e. it has the form(+ + + + − − −−). From this it follows that starting from the Riemannspace with the Lorentz signature (− − −+) and after the extension, weget the additional subspace with local coordinates P,Q,R, S having thesignature (−+++).

Remark. For the Schwarzschild metrics

gij =

⎛⎜⎜⎜⎝− 1

1−m/x 0 0 0

0 −x2 0 00 0 −x2 sin2 y 00 0 0 1− m

x

⎞⎟⎟⎟⎠the Christoffell coefficients are

Γ111 =m

2x(x+m) , Γ122 = −(x+m), Γ133 = −(x+m) sin2 y,

Γ144 = − (x+m)m2x3

, Γ212 =1x , Γ233 = − sin y cos y,

Γ313 =1x , Γ323 =

cos ysin y , Γ414 = − m

2x(x+m) .

The corresponding system (3.1) for the surfaces of translations x(u, v),y(u, v), z(u, v), t(u, v) is nonlinear. After the extension and with the helpof the new coordinates (P,Q,R, S) we get the S8 space with metric ds2

given by

−12ds2 = Γ111Pdx

2 + Γ122Pdy2 + Γ133Pdz

2 + Γ144Pdt2

+ Γ233Qdz2 + 2Γ212Qdxdy + 2Γ

313Rdxdz

+ 2Γ323Rdxdy dz + 2Γ414S dx dt.

7. Anti-Self-Dual Kahler metrics and second order ODE’s

Here we discuss the relations of the equations (1.1) with the theory ofASD-Kahler spaces [27].

It is known that all ASD null Kahler metrics are locally given by

ds2 = −Θttdx2 + 2Θztdxdy −Θzzdy2 + dxdz + dydt

where the function Θ(x, y, z, t) is the solution of the equation

Θxz +Θyt +ΘzzΘtt −Θ2zt = Λ(x, y, z, t),


Λxz + Λyt +ΘttΛzz +ΘzzΛtt − 2ΘztΛtz = 0.

This system of equations has the solution

Θ = −16a1(x, y)z

3 +1

2a1(x, y)z

2t− 1

2a3(x, y)zt

2 +1

6a4(x, y)t

3

which leads to the metric

ds2 = 2(za3− ta4)dx2+4(za2− ta3)dxdy+2(za1− ta2)dy2+2dxdz+2dydtwith geodesics determined by the equation

y′′ + a1(x, y)y′3 + 3a2(x, y)y′2 + 3a3(x, y)y′ + a4(x, y) = 0 (7.1)

In this case the coefficients ai(x, y) are not arbitrary but satisfy theconditions

L1 =∂

∂y(a4y+3a2a4)−

∂

∂x(2a3y−a2x+a1a4)−3a3(2a3y−a2x)−a4a1x = 0,

L2 =∂

∂x(a1x−3a1a3)+

∂

∂y(a3y−2a2x+a1a4)−3a2(a3y−2a2x)+a1a4y = 0.

According to the Liouville theory this means that these equations canbe transformed into the equation

y′′ = 0

by means of point transformations.Notice that the conditions L1 = 0, L2 = 0 are connected with an

integrable nonlinear p.d.e. (as the equation (2.4) for example) and fromthis we can get a lot of examples of ASD–spaces.

8. Dual equations and the Einstein–Weyl geometry in the theoryof second order ODE’s

In the theory of second-order ODE’s

y′′ = f(x, y, y′)

we have the following fundamental diagram:

F (x, y, a, b) = 0↙↗ ↘↖

y′′ = f(x, y, y′) b′′ = g(a, b, b′)

" "

M3(x, y, y′) ⇐⇒ N3(a, b, b′)

144 V. DRYUMA

which shows the relations between a given second-order ODE y′′ = f(x, y, y′),its general integral F (x, y, a, b) = 0 and the so-called dual equation b′′ =g(a, b, b′) which can be obtained from the general integral when the variablesx and y are considered as parameters.

In particular for the equations of type (1.1) the dual equation

b′′ = g(a, b, b′) (8.1)

it follows that the function g(a, b, b′) satisfies the partial differential equa-tion

gaacc + 2cgabcc + 2ggaccc + c2gbbcc + 2cggbccc + g

2gcccc

+ (ga + cgb)gccc − 4gabc − 4cgbbc − cgcgbcc − 3ggbcc (8.2)

− gcgacc + 4gcgbc − 3gbgcc + 6gbb = 0.

According to E. Cartan the expressions for the curvature of the spaceof linear elements (x, y, y′) connected with equation (1.1) is

Ω12 = a[ω

2 ∧ ω21] , Ω01 = b[ω

1 ∧ ω2] , Ω02 = h[ω

1 ∧ ω2] + k[ω2 ∧ ω21 ]

where:

a = −16

∂4f

∂y′4, h =

∂b

∂y′, k = − ∂μ

∂y′− 1

6

∂2f

∂2y′∂3f

∂3y′,

and

6b = fxxy′y′ + 2y′fxyy′y′ + 2ffxy′y′y′ + y′2fyyy′y′ + 2y′ffyy′y′y′

+f2fy′y′y′y′ + (fx + y′fy)fy′y′y′ − 4fxyy′ − 4y′fyyy′ − y′fy′fyy′y′

−3ffyy′y′ − fy′fxy′y′ + 4fy′fyy′ − 3fyfy′y′ + 6fyy .

Two types of equations arise in a natural way: the first type derives fromthe condition a = 0 and the second type from the condition b = 0.

The first condition a = 0 leads to the equation in form (1.1) and thesecond condition leads to the equations (8.1), where the function g(a, b, b

′)

satisfies the above p.d.e..E. Cartan also proved that the Einstein–Weyl 3-manifolds parametrize

the families of curves of equation (8.1) which are dual to the equation (1.1).Some examples of solutions of equation (8.1) were first obtained in [2].For instance if we take

g = a−γA(caγ−1),

we get the equation

[A+ (γ − 1)ξ]2AIV + 3(γ − 2)[A + (γ − 1)ξ]AIII+(2− γ)AIAII + (γ2 − 5γ + 6)AII = 0.


One solution of this equation is

A = (2− γ)[ξ(1 + ξ2) + (1 + ξ2)3/2] + (1− γ)ξ,

which corresponds to the equation

b′′ =1

a[b′(1 + b′2) + (1 + b′2)3/2]

with general integral

F (x, y, a, b) = (y + b)2 + a2 − 2ax = 0

In this case the dual equation has the form

y′′ = − 1

2x(y′3 + y′)

Remark. For more general classes of form-invariant equations the dualequation is introduced in an analogous way.

For example, for the form-invariant equation

Pn(b′)b′′ − Pn+3(b

′) = 0,

where Pn(b′) is a polynomial of degree n in b′ with coefficients depending

on the variables a, b, the dual equation

b′′ = g(a, b, b′)

implies ∣∣∣∣∣∣∣∣∣ψn+4 ψn+3 . . . ψ4ψn+5 ψn+4 . . . ψ5...

......

ψ2n+4 ψ2n+3 . . . ψn+4

∣∣∣∣∣∣∣∣∣ = 0

where the functions ψi are determined with the help of the relations

4!ψ4 = − d2

da2gcc + 4

d

dagbc − gc

(4gbc −

d

dagcc

)+ 3gbgcc − 6gbb,

i ψi =d

daψi−1 − (i− 3)gcψi−1 + (i− 5)gbψi−2, i > 4

For example, the equation

2yy′′ − y′4 − y′2 = 0

with solutionx = a(t+ sin t) + b, y = a(1− cos t)

146 V. DRYUMA

has a dual equation

b′′ = −1atan(b′/2).

According to the above formulas, at n = 1 we get

4!ψ4 =3

2a3tan

c

2

(1 + tan2

c

2

)3,

5!ψ5 = − 15

4a4tan

c

2

(1 + tan2

c

2

)4,

6!ψ6 =90

8a5tan

c

2

(1 + tan2

c

2

)5,

and the relation ∣∣∣∣ ψ5 ψ4ψ6 ψ5

∣∣∣∣ = 0

is satisfied.

We consider next some properties of the Einstein–Weyl spaces [15].A Weyl space is a smooth manifold equipped with a conformal metric

gij(x), and a symmetric connection

Gkij = Γkij −1

2(ωiδ

kj + ωjδ

ki − ωlgklgij)

satisfying a condition of the form

Digkj = ωigkj

where ωi(x) are the components of a given vector field.The Weyl connection Gkij has a curvature tensorW

ijkl and a Ricci tensor

W ijil which is not symmetrical W

ijil �=W i

lij in general.A Weyl space satisfying the Einstein condition

1

2(Wjl +Wlj) = λ(x)gjl(x),

for some function λ(x), is called an Einstein–Weyl space.Let us consider some examples.1. The components of the Weyl connection of the three-dimensional flat

space:ds2 = dx2 + dy2 + dz2

are

2G1 =

⎛⎝ −ω1 −ω2 −ω3ω2 −ω1 0ω3 0 −ω1

⎞⎠ , 2G2 =

⎛⎝ −ω2 ω1 0−ω1 −ω2 −ω30 ω3 −ω2

⎞⎠ ,2G3 =

⎛⎝ −ω3 0 ω10 −ω3 ω2−ω1 −ω2 −ω3

⎞⎠ .


From the equations of the Einstein–Weyl spaces

W[ij] =Wij +Wji

2= λgij

we get the system of equations

ω3x + ω1z + ω1ω3 = 0, ω3y + ω2z + ω2ω3 = 0, ω2x + ω1y + ω1ω2 = 0,

2ω1x + ω2y + ω3z −ω22 + ω

23

2= 2λ, 2ω2y + ω1x + ω3z −

ω21 + ω23

2= 2λ,

2ω3z + ω2y + ω1x −ω21 + ω

22

2= 2λ.

Notice that the first three equations lead to the Chazy equation [16]

R′′′ + 2RR′′ − 3R′2 = 0

for the functionR = R(x+ y + z) = ω1 + ω2 + ω3

where ωi = ωi(x+ y + z).The Einstein–Weyl geometry of the metric gij = diag(1,−eU ,−eU ) and

vector ωi = (2Uz , 0, 0) is determined by the solutions of equation [17]

Uxx + Uyy = (eU )zz.

After substituting U = U(x+ y = τ, z) this equation becomes

Uτ = (eU/2)z.

The consideration of the E–W structure for the metrics

ds2 = dy2 − 4dxdt − 4U(x, y, t)dt2

leads to the dispersionless KP equation [18]

(Ut − UUx)x = Uyy.

2. For the four-dimensional Minkovskii space

ds2 = dx2 + dy2 + dz2 − dt2,the components of the Weyl connection are

2G1 =

⎛⎜⎜⎝−ω1 −ω2 −ω3 −ω4ω2 −ω1 0 0ω3 0 −ω1 0−ω4 0 0 −ω1

⎞⎟⎟⎠ , 2G2 =

⎛⎜⎜⎝−ω2 ω1 0 0−ω1 −ω2 −ω3 −ω40 ω3 −ω2 00 −ω4 0 −ω2

⎞⎟⎟⎠ ,

148 V. DRYUMA

2G3 =

⎛⎜⎜⎝−ω3 0 ω1 00 −ω3 ω2 0−ω1 −ω2 −ω3 −ω40 0 −ω4 −ω3

⎞⎟⎟⎠ , 2G4 =

⎛⎜⎜⎝−ω4 0 0 ω10 −ω4 0 −ω20 0 −ω4 −ω3−ω1 −ω2 −ω3 −ω4

⎞⎟⎟⎠ .The Einstein–Weyl condition

W[ij] =Wij +Wji

2= λgij

where

Wij =Wlilj

and

W kilj =

∂Gkij∂xl

− ∂Gkil

∂xj+GkinG

nlj −GkjnGnil

leads to the system of equations

ω3x + ω1z + ω1ω3 = 0, ω3y + ω2z + ω2ω3 = 0,

ω2x + ω1y + ω1ω2 = 0, ω4x + ω1t + ω1ω4 = 0,

ω4y + ω2t + ω2ω4 = 0, ω4z + ω3t + ω3ω4 = 0,

3ω1x + ω2y + ω3z − ω4t + ω24 − ω22 − ω23 = 2λ,

3ω2y + ω1x + ω3z − ω4t + ω24 − ω21 − ω23 = 2λ,

3ω3z + ω2y + ω1x − ω4t + ω34 − ω21 − ω22 = 2λ,

3ω4t − ω2y − ω1x − ω3z + ω23 + ω21 + ω22 = 2λ.

9. On the solutions of dual equations

The equation (3.2) can be rewritten in the compact form

d2gccda2

− gcdgccda− 4dgbc

da+ 4gcgbc − 3gbgcc + 6gbb = 0 (9.1)

whered

da=∂

∂a+ c

∂

∂b+ g

∂

∂c.

It admits many types of the reductions, the simplest of them being

g = cαω[acα−1], g = cαω[bcα−2], g = cαω[acα−1, bcα−2],g = a−αω[caα−1], g = b1−2αω[cbα−1], g = a−1ω(c− b/a),g = a−3ω[b/a, b− ac], g = aβ/α−2ω[bα/aβ, cα/aβ−α].


The corresponding equations can be integrated in some particular cases.For instance, let us assume that

g = g(a, c) .

From the condition (9.1) we get

d2gccda2

− gcdgccda

= 0 (9.2)

whered

da=∂

∂a+ g

∂

∂c.

By inserting the relation

gac = −ggcc + χ(gc)

in (9.2), we get the following equation for χ(ξ), ξ = gc

χ(χ′′ − 1) + (χ′ − ξ)2 = 0.

It has the solutions

χ =1

2ξ2, χ =

1

3ξ2

So we get two reductions of the equation (9.1) given by

gac + ggcc −g2c2= 0

and

gac + ggcc −g2c3= 0.

Remark. The first reduction of equation (9.1) follows from its represen-tation in the form

gac + ggcc −1

2gc

2 + cgbc − 2gb = h,

hac + ghcc − gchc + chbc − 3hb = 0

and was already considered in [3].In the particular case h = 0 we get

gac + ggcc −1

2g2c + cgbc − 2gb = 0

which is the equation (9.1) for the function g = g(a, c). It can be integratedby means of a Legendre transformation (see [3]).

150 V. DRYUMA

The solutions of the equation

uxy = uuxx + εu2x

were constructed in [19]. In [20] it was showed that they can be written as

u = B′(y) +∫[A(z)− εy](1−ε)/εdz,

x = −B(y) +∫[A(z) − εy]1/εdz.

To integrate the above equations we apply the parametric representation

g = A(a) + U(a, τ), c = B(a) + V (a, τ).

By using the formulas

gc =gτcτ, ga = ga + gτ τa

and after substitution in (9.1), we get the conditions

A(a) =dB

da

and

Uaτ −(VaUτVτ

)τ+ U

(UτVτ

)τ− 1

2

U2τ

Vτ= 0.

So we obtain one equation for two functions U(a, τ) and V (a, τ). Anysolution of this equation determines a solution of equation (9.1).

Let us consider the example

A = B = 0, U = 2τ − aτ2

2, V = aτ − 2 log(τ).

By using the representation

U = τωτ − ω, V = ωτ

it is possible to obtain other solutions of this equation.The problem of integration of the dual equation with a right-hand side

g = g(a, b′) depending on the two variables a and b′ was solved in work[28]. Here we present the construction of such type of solutions.

Proposition 9.1 If h �= 0 and g = g(a, c), the equation (9.1) is equivalentto the equation

Θa

(ΘaΘc

)ccc−Θc

(ΘaΘc

)acc

= 1, (9.3)

where

g = −ΘaΘc, hc =

1

Θc.


To integrate this equation we use the representation

c = Ω(Θ, a).

From the relations

1 = ΩΘΘc, 0 = ΩΘΘa +Ωc

we obtain

Θc =1

ΩΘ, Θa = −

ΩaΩΘ

andΩaΩΘ

(Ωa)ccc +1

ΩΘ(Ωa)cca = 1

Now we get

Ωac =ΩaΘΩΘ

= (log ΩΘ)a = K, Ωacc =KΘ

ΩΘ

Ωaccc =

(KΘ

ΩΘ

)Θ

1

ΩΘ, (Ωacc)a =

(KΘ

ΩΘ

)a− ΩaΩΘ

(KΘ

ΩΘ

)Θ.

As a result the equation (9.3) takes the form[(log ΩΘ)aΘ

ΩΘ

]a= ΩΘ

and can be integrated with the help of the substitution

Ω(Θ, a) = Λa.

So we get the following Abel type equation for the function ΛΘ

ΛΘΘ =1

6Λ3Θ + α(Θ)Λ

2Θ + β(Θ)Λ(Θ) + γ(Θ) (9.4)

with arbitrary coefficients α, β, γ.As an example, let us discuss the case.

α = β = γ = 0 .

The solution of equation (9.4) is

Λ = A(a) − 6√B(a)− 1

3Θ

and we get

c = A′ − 3B′√B − 1

3Θ

152 V. DRYUMA

or

Θ = 3B − 27 B′2

(c −A′)2This solution corresponds to the equation

b′′ = −ΘaΘc

= − 1

18B′b′3 +

A′

6B′b′2 +

(B′′B′− A

′2

6B′)b′ +A′′ +

A′3

18B′− A

′B′′

B′

with a cubic dependence on the first derivative b′ and arbitrary coefficientsA(a), B(a). This equation is equivalent to the equation

b′′ = 0

under application of appropriate point transformations.In fact from the formulas

L1 =∂

∂y(a4y + 3a2a4)−

∂

∂x(2a3y − a2x + a1a4)− 3a3(2a3y − a2x)− a4a1x,

L2 =∂

∂x(a1x − 3a1a3) +

∂

∂y(a3y − 2a2x + a1a4)− 3a2(a3y − 2a2x) + a1a4y,

derived from the components of a projective curvature of the space of linearelements for

y′′ + a1(x, y)y′3+ 3a2(x, y)y

′2 + 3a3(x, y)y′ + a4(x, y) = 0,

we obtain

a1(x, y) =1

18B′, a2(x, y) = −

A′

18B′,

a3(x, y) =A′2

18B′− B′′

3B′, a4(x, y) =

A′B′′

B′− A′3

18B′−A′′,

as well as the conditions

L1 = 0, L2 = 0.

This means that our equation is determined by a projective flat struc-ture in the space of elements (x, y, y′).

Remark. The conditions

L1 = 0, L2 = 0

correspond to the solutions of the equation (1.3) of the form

g(a, b, b′) = A(a, b)b′3 + 3B(a, b)b′2 + 3C(a, b)b′ +D(a, b).


10. Third-order ODE’s and Weyl geometry

The geometry of the equation

b′′′ = g(a, b, b′, b′′)

with a general integral of the form

F (a, b,X, Y, Z) = 0,

was studied by E. Cartan.It was showed that there are a lot of types of geometrical structures

connected with this type of equations.More recently [21, 22] the geometry of third-order ODE’s was considered

in the context of the null-surface formalism and it was discovered that thefunction g(a, b, b′, b′′) satisfies the conditions:

d2grda2

− 2grdgrda− 3dgc

da+4

9g3r + 2gcgr + 6gb = 0, (10.1)

d2grrda2

− dgcrda

+ gbr = 0, (10.2)

whered

da=∂

∂a+ c

∂

∂b+ r

∂

∂c+ g

∂

∂r.

We present here some solutions of the equations (21,22) which areconnected with the theory of second order ODE’s.

Following the notations of E. Cartan we study the third-order differen-tial equations

y′′′ = F (x, y, y′, y′′)

where the function F satisfies the system of equations

d2F2dx2

− 2F2dF2dx− 3dF1

dx+4

9F 32 + 2F1F2 + 6F0 = 0 (10.3)

d2F22dx2

− dF12dx

+ F02 = 0 (10.4)

where Fk ≡ ∂F/∂y(k) andd

dx=∂

∂x+ y′

∂

∂y+ y′′

∂

∂y′+ F

∂

∂y′′.

In particular the third order equation

y′′′ =3y′y′′2

1 + y′2

154 V. DRYUMA

of all cycles on the plane is a good example which turns out to be connectedwith the Einstein–Weyl geometry.

We consider the case

y′′′ = F (x, y′, y′′).

Here F0 = 0 and from the second equation we have

Hx2 + y′′H12 + FH22 = 0,

where

Fx2 + FF22 −F 22

2+ y′′F12 − 2F1 = H.

Taking into account this relation, the first equation gives us the condi-tion

Hx + y′′(H1 − F11)− FF12 −

1

18F 32 − Fx1 = 0.

IfH = H(F2), with F ≡ F (x, y′′)

then we get

Fx2 + FF22 −F 22

3= 0.

The corresponding third-order equation is

y′′′ = F (x, y′′)

and it is connected with the second-order equation

z′′ = g(x, z′).

Another example is the solution of the system for a function F =F (x, y′, y′′) obeying the equation

Fx2 + FF22 −F 22

2+ y′′F12 − 2F1 = 0.

In this case H = 0 and we get the system of equations

Fx2 + FF22 −F 22

2+ y′′F12 − 2F1 = 0,

y′′F11 + FF12 +1

18F 32 + Fx1 = 0,

with the condition of compatibility(F 22

6− F1

)F22 + 2F2F12 + 3F11 = 0.


Acknowledgements

The author would like to thank the Cariplo Foundation (Center Landau–Volta, Como, Italy) as well as the INTAS-99-01782 Programme and theRoyal Swedish Academy of Sciences for financial support.

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2. V. Dryuma, Projective duality in theory of the second order differential equations,Mathematical Researches, Kishinev, Stiintsa, 1990, vol. 112, 93–103.

3. V.S. Dryuma, On Initial values problem in theory of the second order ODE’s, Pro-ceedings of the “Workshop on Nonlinearity, Integrability and all that: Twenty yearsafter NEEDS’79”, Gallipoli(Lecce), Italy, July 1–July 10, 1999, ed. M. Boiti, L.Martina, F. Pempinelli, B. Prinari and G. Soliani. World Scientific, Singapore, 2000,109–116.

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8. A. Tresse, Determination des Invariants ponctuels de l’equation differentielleordinaire de second ordre: y′′ = w(x, y, y′). Preisschriften der furstlichenJablonowski’schen Gesellschaft XXXII, Leipzig, S. Hirzel, 1896.

9. A. Tresse, Sur les invariants differentiels des groupes continus de transformations,Acta Math. 18, 1–88, 1894.

10. E. Cartan, Sur les varietes a connexion projective, Bulletin de la Societe Mathemat.de France 52, 205–241, 1924.

11. G. Thomsen, Uber die topologischen Invarianten der Differentialgleichung y′′ =f(x, y)y′3+ g(x, y)y′2+h(x, y)y′+k(x, y), Abhandlungen aus dem mathematischenSeminar der Hamburgischen Universitat, 7, 301–328, 1930.

12. E.M. Paterson and A.G. Walker, Riemann extensions, Quart. J. Math. Oxford 3,19–28, 1952.

13. E. Wilczynski, Projective Differential Geometry of Curved Surfaces, Transactions ofthe American Mathematical Society, 9, 103–128, 1908.

14. E. Cartan, Sur une classe d’espaces de Weyl, Ann. Ec. Norm. Sup. 14, 1–16, 1943.15. H. Pedersen and K.P. Tod, Three-dimensional Einsten–Weyl Geometry Advances

in Mathematics 97, 71–109, 1993.16. J. Chazy, Sur les equations differentielle dont l’integrale possede une coupure

essentielle mobile, C.R. Acad. Sc. Paris 150, 456–458, 1910.

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17. R. Ward, Einstein–Weyl spaces and Toda fields, Classical and Quantum Gravitation7, L45–L48, 1980.

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qc/0005120, 26 May 2000.23. K.P. Tod, Einstein–Weyl spaces and third-order differential equations, J. Math.

Phys., N9, 2000.24. S. Frittelli, C.N. Kozameh and E.T. Newman, Differential geometry from differential

equations, arXiv:gr-qc/0012058, 15 Dec. 2000.25. J.C. Sprot, Symplest dissipative chaotic flow, Physics Letters A 228, 271–274, 1997.26. B. Konopelchenko, The non-abelian (1+1)-dimensional Toda lattice, Physics Letters

A 156, 221–222, 1991.27. M. Dunajski, Anti-self-dual four-manifolds with a parallel spinor, arXiv:math.

DG/0102225, 18 Oct. 2001.28. V. Dryuma and M. Pavlov. On initial value problem in theory of the second order

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of the second order ODE, Theoretical and Mathematical Physics, 128, N1, 15–26,2001.

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DUNKL OPERATORS AND

CALOGERO–SUTHERLAND MODELS

F. FINKEL, D. GOMEZ-ULLATE, A. GONZALEZ-LOPEZ,M.A. RODRIGUEZ and R. ZHDANOVDpto. de Fısica Teorica II, Universidad Complutense, 28040Madrid, Spain

Abstract. We describe a general method for constructing (scalar or spin) Calogero–Sutherland models of AN or BCN type, which are either exactly or quasi-exactly solvable.Our approach is based on the simultaneous use of three different families of Dunkloperators of each type, one of which was recently introduced by the authors. We performa complete classification of the models which can be constructed by our method. Weobtain in this way several new families of (quasi-)exactly solvable Calogero–Sutherlandmodels, some of them with elliptic interactions.

1. Introduction

The quantum Calogero [6] and Sutherland [61, 62] models describe a systemof N particles on the line and the circle with pairwise interactions falling offas the inverse square of the distance between them. Both models are charac-terized by two key mathematical properties, namely complete integrabilityand exact solvability , which have probably been one of the main motivationsfor the vast attention devoted to these models since their introduction inthe early seventies. The physical and mathematical interest of these modelsis reflected by their appearance in such diverse areas as fractional statisticsand anyons [51, 5, 30, 31, 7, 41], quantum Hall liquids [1], Yang–Millstheories [29, 12], random matrix theory [64, 37], integrable PDEs withsoliton solutions [43, 55], and orthogonal polynomials [44, 2, 13, 16].

A landmark in the development of this field was in fact Olshanetskyand Perelomov’s proof of the complete integrability of the Calogero andSutherland models [48]. In their construction, the integrability of the modelsfollows by expressing the Hamiltonian as one of the radial parts of theLaplace–Beltrami operator in a symmetric space associated with the ANroot system. Furthermore, these authors also constructed new families of

157


158 F. FINKEL ET AL.

integrable many-body Hamiltonians associated with other root systems,and showed that the most general interaction potential for these models isproportional to the Weierstrass ℘ function. We shall use in what followsthe customary term Calogero–Sutherland (CS) to collectively refer to thesemodels.

As previously mentioned, the original models introduced by Calogeroand Sutherland are exactly solvable (ES) in the sense that the Hamiltonianpreserves a known infinite increasing sequence (or flag) of finite-dimensionalsubspaces of smooth functions [57, 66]. This property is shared by some butnot all the CS models introduced in Olshanetsky and Perelomov’s paper.Some of these models —such as the elliptic model of BCN type [38, 23]—are in fact quasi-exactly solvable (QES), which means that the Hamiltonianleaves invariant some known finite-dimensional functional space, so thata subset of the spectrum can be computed by diagonalizing the finite-dimensional matrix of the restriction of the Hamiltonian to this space,see [65, 59, 67, 27, 49] and references therein. In contrast, the possible(quasi-)exact solvability of the elliptic model of AN type remains an openquestion.

Quasi-exactly solvable models are typically constructed by a two-stepprocedure: i) take a quadratic combination of the generators of an algebra offirst-order differential operators with a finite-dimensional module of smoothfunctions; ii) solve the equivalence problem, that is, determine whetherthis second-order operator can be reduced to the Schrodinger form by atransformation preserving the QES character. This procedure —sometimescalled the hidden symmetry algebra approach— is most easily implementedin the case of a spinless particle in one spatial dimension. In this case theonly relevant algebra is the realization of the Lie algebra sl(2) spannedby {∂z , z∂z − m

2 , z2∂z − mz}, which for integer m preserve the space of

polynomials in z of degree at most m [42]. Moreover, every second degreepolynomial in these generators is equivalent to a Schrodinger operator bya suitable change of the independent variable and a gauge transformation.This basic property enabled Gonzalez-Lopez, Kamran and Olver [25, 27] toperform a complete classification of the resulting QES potentials.

The extension of this construction to several variables (with or withoutinternal degrees of freedom) is considerably more involved. In the first place,there are many algebras of first-order differential operators admitting aninvariant finite-dimensional functional space. For instance, in two variablesthere are exactly 17 inequivalent families of finite-dimensional Lie algebrasof first-order differential operators (plus 5 additional real forms) with aninvariant module, [24, 28]. (No such classification in more than two vari-ables has been completed so far.) Secondly, the corresponding equivalenceproblem cannot be explicitly solved for N > 1 variables [11, 17]. This

SPIN CALOGERO–SUTHERLAND MODELS 159

explains why even in two variables there is no exhaustive classification ofQES models, but rather an extensive collection of examples [60, 26, 70].In N variables, some QES deformations of the CS models in Olshanetskyand Perelomov’s paper have been recently constructed using a realizationof sl(N + 1) [47, 36, 22] as the hidden symmetry algebra.

A good deal of effort has been devoted over the past decade to the studyCalogero–Sutherland models with spin, due in part to their intimate con-nection with integrable spin chains of Haldane–Shastry type [33, 58]. SpinCS models have been explored by several techniques, such as the exchangeoperator formalism [52, 46], the Dunkl operators approach [4, 10, 16], reduc-tion by discrete symmetries [56], and construction of Lax pairs [34, 35, 40].Historically, the first CS spin models discussed in the literature were thetrigonometric [32] and rational [46] spin versions of the original Sutherlandand Calogero models of AN type. The exact solvability and integrability ofboth models can be established by relating the Hamiltonian to a quadraticcombination of either the Dunkl [15] or the Dunkl–Cherednik [9, 10] op-erators of AN type, respectively. To the best of our knowledge, the onlyadditional spin CS model of AN type discussed in the literature prior to2001 was the hyperbolic Sutherland model with an external Morse po-tential [39]. In the BCN case, only the rational and trigonometric spinmodels had been constructed [68] prior to that date. In the latter paper, thespectrum of the rational BCN spin model was explicitly determined, and itsintegrability was shown by means of the Lax pair approach. An alternativeproof of the integrability of this model using Dunkl operators of BN typewas later presented in [69]. The same operators were later employed byDunkl to construct a complete basis of eigenfunctions [16]. In contrast, thesolvability and/or integrability of the trigonometric/hyperbolic BCN modelwas still unproved by 2001.

In this paper, we review the systematic method for constructing Calogero–Sutherland models with spin using Dunkl operators introduced in Refs. [19,20]. Our construction is reminiscent of the hidden symmetry algebra ap-proach to (Q)ES scalar models, in the sense that it also involves generalquadratic combinations of different (families of) Dunkl operators, whichleave invariant a common polynomial space of finite dimension. Our con-tributions can be summarized as follows:

1. We have introduced a third new family of Dunkl operators (in theirAN and BN versions), which, together with the two previously knownfamilies leave invariant a finite-dimensional polynomial subspace.

2. We have shown that certain quadratic combinations involving the threefamilies of Dunkl operators of either AN or BN type are equivalent toa Schrodinger operator.

3. We have performed a complete classification of the (Q)ES spin CS mod-


els of AN and BCN types which can be constructed by this method.All the previously known families of spin CS models appear in theclassification. Several families of quantum spin CS models with ellipticinteractions are obtained for the first time.

4. In the AN case, we have shown that each spin CS model yields acorresponding scalar one, which is (Q)ES by construction. Moreover,we have shown that each one-body (Q)ES potential in the Gonzalez-Lopez, Kamran and Olver’s classification [25, 26] is associated with a(Q)ES Calogero–Sutherland model (with or without spin).

5. In the BCN case, we have proved that all the potentials in the classi-fication are expressible in a unified way in terms of the Weierstrass ℘function with suitable (sometimes infinite) half-periods. This providesa natural spin counterpart of Olshanetsky and Perelomov’s formula fora general scalar potential related to the BCN root system.

The rest of the paper is organized as follows. In Section 2 we presentthe results for the AN root system. To this end, we first introduce therelevant families of Dunkl operators and discuss their properties. We thenexplain the details of the construction of the many-body Hamiltonians.Finally, we describe the classification of the models. Section 3 is devotedto the parallel construction of the models of BCN type. We point out themain differences with respect to the AN case and derive the general formulafor the potentials in terms of the Weierstrass ℘ function. In Section 4, wepresent some concluding remarks and related open questions. We refer thereader to Refs. [19, 20] for a complete account of the intermediate resultsand proofs not included in this review.

2. Calogero–Sutherland spin models of AN type

2.1. A NEW FAMILY OF DUNKL OPERATORS OF AN TYPE

The Dunkl operators

J−i =∂

∂zi+ a

∑j �=i

1

zi − zj(1−Kij), i = 1, . . . , N, (2.1)

were introduced in Ref. [15] in connection with the theory of orthogonalpolynomials in several variables. In the latter expression, a is a real pa-rameter and the sum runs over 1, . . . , i − 1, i + 1, . . . , N . In general, allsums and products in this paper will range from 1 to N unless otherwiserestricted. The operators Kij = Kji are coordinate permutation operators,whose action on a function f(z), z = (z1, . . . , zN ) ∈ IRN , is given by

(Kijf) (z1, . . . , zi, . . . , zj , . . . , zN ) = f(z1, . . . , zj , . . . , zi, . . . , zN ). (2.2)


The Dunkl operators (2.1) form a commuting family, and together withthe permutation operators Kij span a degenerate affine Hecke algebra [10].These properties play an essential role in the proof of the integrabilityand the computation of the spectrum of the rational spin model of ANtype [4, 3]. The very same properties are also satisfied by the operators

J0i = zi∂

∂zi+ a

∑j<i

zizi − zj

(1−Kij)+ a∑j>i

zjzi − zj

(1−Kij)+ 1− i, (2.3)

i = 1, . . . , N , introduced by Cherednik in connection with the trigonometricmodel of AN type. We shall employ the slightly more symmetric (but non-commuting) form of the Cherednik operators given by1

J0i = zi∂

∂zi− m2+a

2

∑j �=i

zi + zjzi − zj

(1−Kij), i = 1, . . . , N , (2.4)

where m is a nonnegative integer whose significance will become clearshortly.

The third new family of Dunkl operators of AN type introduced by theauthors in [19] reads

J+i = z2i∂

∂zi−mzi + a

∑j �=i

zizjzi − zj

(1−Kij), i = 1, . . . , N. (2.5)

These operators also form a commuting family and span a degenerate affineHecke algebra. As far as the solvability of the resulting models is concerned,the important commutation relations are

[Kij , Jεk] = 0 , KijJ

εi = J

εjKij , (2.6)

where ε = ±, 0, and the indices i, j, k are all different.The key property in our construction of (quasi-)exactly solvable spin

CS models is the fact that the operators Jεi preserve a common polynomialsubspace of finite-dimension. Indeed, consider the polynomial spaces

Rn(z) = 〈zl11 · · · zlNN | l1 ≤ n, . . . , lN ≤ n〉 , (2.7)

Tn(z) = 〈zl11 · · · zlNN | l1 + · · · + lN ≤ n〉 , (2.8)

which shall be referred to as the rectangular and triangular modules, re-spectively, by analogy with the two variable case [18].

1 Note that the operators J0i and J0i differ by a linear combination with constantcoefficients of the permutation operators Kij . Therefore, the operators J

0i and Kij also

span a degenerate affine Hecke algebra.


Theorem 2.1 The operators J−i and J0i preserve the modules Tn(z) andRn(z) for any nonnegative integer n. The operators J+i preserve the moduleRm(z), but do not preserve the modules Tn(z) and Rn(z) for n �= m.

Corollary 2.2 Any polynomial in the operators Jεi preserves the rectangu-lar module Rm(z). In addition, if the polynomial does not depend on J+i ,it preserves the modules Rn(z) and Tn(z) for all n.

2.2. CONSTRUCTION OF SPIN CS MODELS OF AN TYPE

Let S be the Hilbert space of the particles’ internal degrees of freedom or“spin”, given explicitly by

S = span{ |s1, . . . , sN 〉}−M≤si≤M , M ∈ 12IN .

Consider the spin permutation operators Sij, i, j = 1, . . . , N , whose actionon a spin state |s1, . . . , sN 〉 is given by

Sij |s1, . . . , si, . . . , sj , . . . , sN 〉 = |s1, . . . , sj , . . . , si, . . . , sN 〉 . (2.9)

The operators Sij are represented in S by (2M + 1)N -dimensional sym-metric matrices. The permutation operators Sij can be easily expressedin terms of one-particle spin operators. For instance, if M = 1/2 we haveSij = 2σi ·σj+ 1

2 , where σi ≡ (σ1i , σ2i , σ3i ) are the usual SU(2) spin operatorsassociated with the i-th particle.

The starting point in our construction is the following quadratic com-bination involving all three families of Dunkl operators of AN type:

−H =∑i

(c++(J

+i )

2 + c00(J0i )

2 + c−−(J−i )2 +

c0+2{J0i , J+i }

+c0−2{J0i , J−i }+ c+J+i + c0J0i + c−J−i

),

(2.10)

where cεε′ , cε , ε, ε′ = ±, 0 , are arbitrary real constants. The operator H

possesses the following obvious properties:

i) It preserves Rm (and also Rn and Tn for all n if c++ = c0+ = c+ = 0).ii) It commutes with Kij for all values of i, j.iii) It is completely determined by the pair of polynomials (P,Q) given by

P (z) = c++z4 + c0+z

3 + c00z2 + c0−z + c−− ,

Q(z) = c+z2 + c0z + c− .

(2.11)

Let Λ be the projector on states antisymmetric under the simultaneousinterchange of any two particles’ coordinates and spins. In terms of the


total permutation operators Πij ≡ KijSij, the projector Λ is defined by therelations Λ2 = Λ and ΠijΛ = −Λ, j > i = 1, . . . , N . Since K2

ij = 1, thelatter relations are equivalent to

KijΛ = −SijΛ, j > i = 1, . . . , N. (2.12)

The antisymmetrizer Λ is given by a polynomial in the total permutationoperators Πij . For instance, for N = 1, 2 we have:

N = 2 : Λ =1

2(1−Π12) ,

N = 3 : Λ =1

6(1−Π12 −Π13 −Π23 +Π12Π13 +Π12Π23) .

Since H commutes with Kij and acts trivially on S, it follows that Hcommutes with Λ and therefore preserves Λ(Rm ⊗S).

Given a scalar differential operator D, define the linear mapping ∗ bythe requirement that

(DKi1j1 · · ·Kirjr)∗ = (−1)rDSirjr · · ·Si1j1 . (2.13)

It follows from this definition and Eq. (2.12) that

HΛ = H∗Λ ,

and thus H∗also preserves Λ(Rm ⊗S).

Inserting the expressions (2.1), (2.4) and (2.5) into Eq. (2.10), aftersome algebra we obtain

−H∗ =∑i

(P (zi)∂

2zi + Q(zi)∂zi +R(zi)

)+ ac++(1−m)

∑i�=jzizj

+ 2a∑i�=j

P (zi)

z−ij∂zi +

a

2

∑i�=j

(c++(z

+ij )

2 + c0+z+ij + c00

)Sij

− a∑i�=j

P (zi)

(z−ij)2(1 + Sij) +

a2

12c00

∑i,j,k

′(1− SijSik),

where P and Q are given in Eq. (2.11),

Q(z) = Q(z) +(1− �

2

)P ′(z) , � = 1 +m+ a(N − 1) ,

R(z) = c++(�+m(m− 2)− 1)z2+[c0+

[m(m− 1− �

2

)+ �− 1

]−mc+

]z + c00

4 (2(�− 1) +m(m− 2)) − m2 c0 ,

(2.14)

z±ij ≡ zi ± zj and∑i,j,k

′ denotes summation in i, j, k with i �= j �= k �= i.

Note that the terms in first derivatives do not depend on the spin per-mutation operators Sij . This fact is essential for the existence of a gauge


transformation and a change of the independent variables mapping H∗into

a Schrodinger operator, i.e.,

μ(z) ·H∗(z) · μ−1(z)∣∣∣z=ζ(x)

≡ H = −∑i

∂2xi + V (x) , (2.15)

where V (x) is a Hermitian matrix-valued function.

Theorem 2.3 The operator H∗can be reduced to a Schrodinger operator

H by the change of variables

xi = ζ−1(zi) =

∫ zi dy√P (y)

, i = 1, . . . , N, (2.16)

and the gauge transformation with gauge factor

μ(z) =∏i<j

|z−ij |a∏i

P (zi)− 1

4 exp

∫ zi Q(y)

2P (y)dy. (2.17)

The potential is given by

V (x) = a2∑i,j,k

′ P (zi)z−ij z

−ik

+ a∑i�=j

Q(zi)

z−ij+ a

∑i�=j

P (zi)

(z−ij )2(a+ Sij)

− a2

∑i�=j

(c++(z

+ij )

2 + c0+z+ij + c00

)Sij −

a2

12c00

∑i,j,k

′(1− SijSik)

− ac++(1−m)∑i�=jzizj +

1

4

∑i

W (zi)

∣∣∣∣∣z=ζ(x)

,

(2.18)where

W (z) =1

P (z)

(3

4P ′(z)2 + Q(z)2 − 2Q(z)P ′(z)

)− P ′′(z) + 2Q′(z)− 4R(z).

Corollary 2.4 The Schrodinger operator (2.15) with potential (2.18) leavesinvariant the finite-dimensional space

Rm = μ(ζ(x))Λ(Rm(ζ(x))⊗S

). (2.19)

In addition, if c++ = c0+ = c+ = 0, it preserves the space Rn and

Tn = μ(ζ(x))Λ(Tn(ζ(x))⊗S

), (2.20)

for any nonnegative integer n.


Remark 2.5. The change of variables (2.16) is defined up to an arbitrarytranslation for each coordinate xi, i = 1, . . . , N . This arbitrariness can beremoved in some cases by requiring that the resulting potential be invariantunder sign reversals of any coordinate xi.

2.3. CLASSIFICATION OF THE MODELS OF AN TYPE

We have seen that any quadratic combination (2.10) yields a (Q)ES Schrodingeroperator or Hamiltonian of the form (2.15), which is totally determined(up to coordinate translations) by the pair (P,Q). However, different pairs(P,Q) may give rise to the same potential. This is a consequence of theinvariance of the vector spaces 〈J−i , J0i , J+i 〉, i = 1, . . . , N , under the pro-jective action of GL(2, IR). The image of the operator H under this action isstill of the form (2.10), albeit with different coefficients cεε′ and cε. We shallmake use of this fact to completely classify the resulting spin CS models.

Consider the mapping

Jεi (w) $→ Jεi (z) = μm(z) · Jεi (w(z)) · μ−1m (z) , (2.21)

wherew = (w1, . . . , wN ) is given by the projective (Mobius) transformation

wi =αzi + β

γzi + δ, i = 1, . . . , N, Δ = αδ − βγ �= 0, (2.22)

and the gauge factor μm(z) is defined by

μm(z) =N∏i=1

(γ zi + δ)m. (2.23)

Lemma 2.6 The mapping (2.21)–(2.23) acts linearly on the vector spaces〈J−i , J0i , J+i 〉, i = 1, . . . , N , as⎛⎜⎜⎜⎝

J+i (z)

J0i (z)

J−i (z)

⎞⎟⎟⎟⎠ =1

Δ

⎛⎜⎜⎜⎝α2 2αβ β2

αγ αδ + βγ βδ

γ2 2γδ δ2

⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝J+i (z)

J0i (z)

J−i (z)

⎞⎟⎟⎟⎠ . (2.24)

Remark 2.7. The irreducible multiplier representation ρn,i of GL(2, IR)on the space of univariate polynomials of degree at most n is defined [50]by the linear transformations

p(z) $→ p(z) = Δi(γz + δ)n p

(αz + β

γz + δ

).

Thus 〈J−i , J0i , J+i 〉, i = 1, . . . , N , is a carrier space of the representationρ2,−1.


It follows from the previous lemma that the operator H defined by

H(z) = μm(z) ·H(w(z)) · μ−1m (z)

is also a second degree polynomial in the Dunkl operators Jεi (z). In fact, it

may be easily shown that the operator H is determined by the transformof the pair (P,Q) under the representation ρ4,−2 ⊕ ρ2,−1.

Using the Mobius transformations (2.21)–(2.23), together with the com-plex projective (linear) transformation wj = i zj , j = 1, . . . , N , the polyno-mial P can be conveniently chosen as one of the following canonical forms(see [19] for more details):

1) 1 ,

2) z ,

3) ν(z2 − 1) ,4) ν(1− z2) ,5) νz2 ,

6) ν(1 + z2)2 ,

7) ν(1− z2)(1− k2z2) ,8) ν(z2 − 1)(1− k′2z2) ,9) ν(1− z2)(k′2 + k2z2) ,

where ν > 0, 0 < k < 1 and k′2 = 1− k2. Each canonical form yields a cor-responding family of (Q)ES spin potentials. Taking advantage of the arbi-trariness mentioned in Remark 2.5 and dropping some inessential constantterms, the resulting potentials can be written in all cases as

V (x) =∑i

U(xi) + Vint(x) , (2.25)

where U represents a scalar external field, and the interaction potential Vintis of the form

Vint(x) =∑i�=j

(V −(x−ij) + V

+(x+ij))a(a+ Sij), x±ij ≡ xi ± xj, (2.26)

with either V + = 0 or V + = V −.

We shall now present the list of potentials for a suitably chosen value ofthe parameter ν. This is justified, since the scaling (cεε′ , cε) $→ (λcεε′ , λcε)induces the mapping

V (x ; cεε′ , cε) $→ V (x ;λ cεε′ , λ cε) = λV (√λx ; cεε′ , cε) (2.27)

of the corresponding potentials. The explicit expressions of the coefficientsA, B, C and D are given in Table I.

Case 1. P (z) =1

4.

External potential :

U(x) = Ax4 +B x3 + C x2 +Dx .


Interaction potential :

Vint(x) =∑i�=j(x−ij)

−2 a(a+ Sij) .

Case 2. P (z) = 4z .External potential :

U(x) = Ax6 +B x4 + C x2 +D

x2.


Vint(x) =∑i�=j

((x−ij)

−2 + (x+ij)−2) a(a+ Sij) .

Case 3. P (z) = 4(z2 − 1) .External potential :

U(x) = A cosh2 2x+B cosh 2x+ C cosh 2x sinh−22x+D sinh−22x .


Vint(x) =∑i�=j

(sinh−2x−ij + sinh

−2x+ij)a(a+ Sij) .

Case 4. P (z) = 4(1− z2) .External potential :

U(x) = A cos2 2x+B cos 2x+ C cos 2x sin−22x+D sin−22x .


Vint(x) =∑i�=j

(sin−2x−ij + sin

−2x+ij)a(a+ Sij) .

Case 5. P (z) = 4z2 .External potential :

U(x) = A e4x +B e2x + C e−2x +D e−4x .


Vint(x) =∑i�=j

sinh−2x−ij a(a+ Sij) .


Case 6. P (z) = (1 + z2)2 .External potential :

U(x) = A cos 4x+B cos 2x+ C sin 4x+D sin 2x .


Vint(x) =∑i�=j

sin−2x−ij a(a+ Sij) .

Case 7. P (z) = 4(1− z2)(1− k2z2) .Here (and also in Cases 8 and 9) the functions snx ≡ sn(x|k), cnx ≡cn(x|k), and dnx ≡ dn(x|k) are the usual Jacobian elliptic functions ofmodulus k, and k′ =

√1− k2 is the complementary modulus.

External potential :

U(x) = A sn22x+B cn 2xdn 2x+ sn−22x (C +D cn 2xdn 2x) .


Vint(x) =∑i�=j

(cn2x−ij dn

2x−ijsn2x−ij

+cn2x+ij dn

2x+ij

sn2x+ij

)a(a+ Sij) .

Case 8. P (z) = 4(z2 − 1)(1 − k′2z2) .External potential :

U(x) = sn−22x (A+B dn 2x) + cn−22x (C +D dn 2x) .


Vint(x) =∑i�=j

(dn2x−ij

sn2x−ij cn2x−ij

+dn2x+ij

sn2x+ij cn2x+ij

)a(a+ Sij) .

Case 9. P (z) = 4(1− z2)(k′2 + k2z2) .External potential :

U(x) = dn−22x (A +B cn 2x) + sn−22x (C +D cn 2x) .


Vint(x) =∑i�=j

(cn2x−ij

sn2x−ij dn2x−ij

+cn2x+ij

sn2x+ij dn2x+ij

)a(a+ Sij) .


Remark 2.8. Some of the potentials that appear in this classification havebeen previously studied by other authors. Case 1 with A = B = 0 yields therational Calogero AN spin model. Case 5 with A = B = 0 or C = D = 0 isthe model studied by Inozemtsev [39], while for A = B = C = D = 0 thehyperbolic Sutherland AN spin model is obtained. Case 6 with A = B =C = D = 0 is the trigonometric Sutherland AN spin model. The remainingpotentials were first introduced in Ref. [19].

Remark 2.9. In Cases 1–5, the potential is ES if A = B = 0. The onlyES potential in Case 6 is the trigonometric Sutherland potential (A = B =C = D = 0). The remaining potentials, including all the elliptic potentialsin Cases 7–9, are QES.

Remark 2.10. We emphasize that the spin potentials in the above listreduce to (Q)ES scalar many-body potentials by the substitution Sij → −1.This is a straightforward consequence of the fact that the operator Hpreserves the space Sm(z) of symmetric polynomials in z of degree at mostm, where the permutation operators Kij act as the identity. This impliesthat the scalar Hamiltonian

H|Sij→−1 = μ(z) ·H(z)|Kij→1 · μ−1(z)∣∣∣z=ζ(x)

preserves the space μ(ζ(x))Sm(ζ(x)). The list of potentials thus obtainedincludes all the previously known examples of scalar (Q)ES CS models.

Remark 2.11. All the potentials in the classification diverge on the hyper-planes xi = xj, 1 ≤ i < j ≤ N , as (x−ij)−2. In some cases there are additionalinverse-square singularities on other hyperplanes. In order to qualify asphysical wavefunctions, the eigenfunctions of H in Rm must vanish atall these hyperplanes faster than the square root of the distance to thehyperplane. In addition, in Cases 1–3 and 5 the eigenfunctions are requiredto be square-integrable over a suitable domain of IRN . Both requirementsimpose certain constraints on the parameters of the potential (see Ref.[25] for a complete treatment of this problem in the one-particle case). Forexample, in Case 5 the change of variable and gauge factor are respectivelygiven by

z = e2x, μ(x) =∏i<j

| sinhx−ij |a∏i

e(c04−m)xi exp

[1

8

(c+e

2xi − c−e−2xi)].

It follows that the spin functions ψ ∈ Rm vanish at the singularities x−ij =0 of the potential faster than |x−ij |1/2 and are square-integrable over thedomain {x ∈ IRN : x1 > · · · > xN} provided a > 1/2, α+ < 0 and α− > 0.


Remark 2.12. The Hamiltonian H describes a system of identical parti-cles, whose corresponding states must thus be either completely symmetricor antisymmetric under particle permutations. Assume for definiteness thatwe are dealing with a system of fermions2. By construction, the elementsof Rm are completely antisymmetric and hence qualify as physical statesprovided the parameters of the potential satisfy certain constraints (seeRemark 2.11). We can also construct completely antisymmetric states of theHamiltonian H in the following alternative way. Consider the symmetrizerΛ defined by the relations Λ2 = Λ and ΠijΛ = Λ. (Re)define the starmapping substituting Eq. (2.13) with the requirement that

(DKi1j1 · · ·Kirjr)∗ = DSirjr · · ·Si1j1 , (2.28)

whereD is a scalar differential operator. Consider the gauge factor μ definedas the antisymmetric extension to IRN of the gauge factor μ restricted tothe domain Ω ≡ {z ∈ IRN : z1 > · · · > zN}. In other words,

μ(z; a) = (−1)ε(i1,...,iN )μ(z; a) ,

where μ(z; a) is given in Eq. (2.23), (i1, . . . , iN ) ∈ SN is the permutationsuch that (zi1 , . . . , ziN ) ∈ Ω, and ε is the parity. We observe that thehamiltonian H is the image of the operator H(−a) (with H(a) given byEq. (2.10)) under the new star mapping (2.28), the gauge transformationwith gauge factor μ(z;−a), and the change of variables (2.16), namely,

H = μ(z;−a) ·H∗(z;−a) · μ−1(z;−a)∣∣∣z=ζ(x)

.

It follows that H preserves the finite-dimensional space of completely anti-symmetric spin functions

Rm = μ(ζ(x);−a) Λ(Rm(ζ(x))⊗S

). (2.29)

Note that the Hamiltonian H may not possess physical wavefunctions inboth spaces Rm and Rm at the same time. For instance, in Case 5, the func-tions in Rm do not vanish in general faster than |x−ij |1/2 at the singularitiesx−ij = 0 of the potential if a > 1/2 (see Remark 2.11).

2 This assumption covers the physically most interesting case for which the internaldegrees of freedom are naturally interpreted as the particles’ spin.


3. Calogero–Sutherland spin models of BCN type

3.1. DUNKL OPERATORS OF BN TYPE

In the BCN case, our construction makes use of the following three familiesof Dunkl operators3:

J−i = ∂zi + a

(∑j �=i

1

z−ij(1−Kij) +

∑j �=i

1

z+ij(1− Kij)

)+b

zi(1−Ki) ,

J0i = zi∂zi −m

2+a

2

(∑j �=i

z+ij

z−ij(1−Kij) +

∑j �=i

z−ijz+ij

(1− Kij)), (3.1)

J+i = z2i ∂zi −mzi + a(∑j �=i

zizj

z−ij(1−Kij)−

∑j �=i

zizj

z+ij(1− Kij)

)− b′zi(1− (−1)mKi),

where i = 1, . . . , N , a, b, b′ are real parameters, Kij ≡ KiKjKij, and Ki aresign-reversing operators. The action of the latter operators on a functionf(z) is given by

(Kif)(z1, . . . , zi, . . . , zN ) = f(z1, . . . ,−zi, . . . , zN ) .

We note that the operators Kij and Ki span the Weyl group of type BN .The first two families in Eq (3.1) were introduced by Dunkl in Ref. [16],while the third one was defined by the authors in Ref. [19]. Each family,together with the operators Kij and Ki span a degenerate affine Heckealgebra. They also satisfy the commutation relations (2.6), along with

[Ki, Jεj ] = 0 , KiJ

εi = (−1)εJεiKi , (3.2)

where i �= j. Just as in the AN case, the key property regarding the (quasi)-exact solvability of the resulting models is the invariance of the module Rmunder these operators. In fact, Theorem 2.1 and its Corollary 2.2 are bothsatisfied by the BN -type Dunkl operators (3.1).

3.2. CONSTRUCTION OF SPIN CS MODELS OF BCN TYPE

In the BN case, we shall use the spin permutation operators Sij and thespin reversing operators Si, whose action on a spin state |s1, . . . , sN 〉 isgiven by

Si|s1, . . . , si, . . . , sN 〉 = |s1, . . . ,−si, . . . , sN 〉 .3 No confusion should arise by the fact both the AN and BN -type Dunkl operators are

denoted as Jεi , ε = ±, 0.


We shall also employ the customary notation Sij = SiSjSij.In theBCN case, the outset of our construction is the following quadratic

combination of the BN -type Dunkl operators (3.1):

−H =∑i

(c++(J

+i )

2 + c00(J0i )

2 + c−−(J−i )2 + c0 J

0i

). (3.3)

The terms∑i{J±i , J0i } and

∑i J±i which appear in theAN case (see Eq. (2.10))

have now been discarded in order to ensure that H commutes with both thepermutation and sign-reversing operators Ki. Note also that H preservesthe spaceRm(z) (or the spacesRn(z) and Tn(z) for arbitrary n, if c++ = 0).

Let Λ′ be the total antisymmetrizer with respect to permutations andsign-reversals of the particles’ coordinates and spins, which is determinedby the relations

Λ′2 = Λ′, KijΛ′ = −SijΛ′, KiΛ

′ = −SiΛ′, (3.4)

for j > i = 1, . . . , N . It may be easily shown that

Λ′ =Λ

2N

∏i

(1−Πi) ,

where Πi ≡ KiSi. Since H commutes with Kij and Ki and acts triviallyin S, it commutes with Λ′ and thus preserves Λ′(Rm ⊗S). We now definethe linear mapping ∗ by the condition

(DKα1 · · ·Kαr)∗ = (−1)rDSαr · · ·Sα1 , (3.5)

where D is a scalar differential operator and the subindex α stands eitherfor ij or i. Just as before, this definition and Eq. (3.4) imply that HΛ′ =H∗Λ′. It follows that H∗ also preserves Λ′(Rm ⊗ S) (or Λ′(Rn ⊗ S) and

Λ′(Tn ⊗S) for arbitrary n, if c++ = 0).Inserting the expressions of the Dunkl operators (3.1) into (3.3), one

obtains

−H∗ =∑i

(P (zi)∂

2zi + Q(zi)∂zi +R(zi)

)+ 4a

∑i�=j

ziP (zi)

z2i − z2j∂zi

−∑i

(b c−−z2i

(1 + Si) + b′c++z

2i (1 + (−1)mSi)

)− a

∑i�=jP (zi)

(1 + Sij

(z−ij)2+1 + Sij

(z+ij)2

)

+a c++

2

∑i�=j

((z+ij )

2(1 + Sij) + (z−ij )

2(1 + Sij))+ C ,

(3.6)


where

P (z) = c++z4 + c00z

2 + c−− ,

Q(z) = 2c++(1−m− b′ + 2a(1 −N))z3

+(c0 + c00(1−m+ 2a(1 −N))

)z +

2bc−−z

,

R(z) = c++m(m− 1 + 2b′)z2 ,

C = c00

[Nm2

4+a2

12

(∑i,j,k

′[4− (Sij + Sij)(Sik + Sik)]

+ 6∑i�=j(1− SiSj)

)+a

2

∑i�=j(2 + Sij + Sij)

]− Nmc0

2.

We observe that the first-derivatives terms of H∗do not involve the spin

operators, which otherwise makes it very difficult (if not impossible) to finda gauge transformation reducing H

∗to a Schrodinger operator4. The exis-

tence of such gauge factor is one of the main ingredients in our construction.

Theorem 3.1 The operator H∗in Eq. (3.6) can be reduced to a Schroding-

er operator H by the change of variables

xi = ζ−1(zi) =

∫ zi dy√P (y)

, i = 1, . . . , N, (3.7)

and the gauge transformation generated by

μ(z) =∏i<j

|z2i − z2j |a ∏

i

P (zi)− 1

4 exp

∫ zi Q(yi)

2P (yi)dyi . (3.8)

The corresponding potential is

V (x) = a∑i�=jP (zi)

(a+ Sij

(z−ij )2+a+ Sij

(z+ij)2

)

− c++

2

((z+ij )

2(1 + Sij) + (z−ij)

2(1 + Sij))

+∑i

(b′c++ z

2i (1 + (−1)mSi) +

b c−−z2i

(1 + Si) +W (zi)

)

+ aN(N − 1)[c0 −

c003(a(2N − 1) + 3(m− 1))

]− C

∣∣∣∣∣z=ζ(x)

,

(3.9)

4 For this reason, we have omitted the term∑

i[J+i , J

−i ] in (3.3).


where

W (z) =1

2

(Q′ − P

′′

2

)+

1

4P

(Q− P

′

2

)(Q− 3P ′

2

)+ c z2 ,

and

c = − c++

(2a2(N − 1)(2N − 1) + 4a(N − 1)(b′ +m− 1)

+m(2b′ +m− 1)).

Corollary 3.2 The Schrodinger operator (2.15) with potential (3.9) leavesinvariant the finite-dimensional space

R′

m= μ(ζ(x))Λ′

(Rm(ζ(x))⊗S

)(3.10)

In addition, if c++ = 0, it preserves the space R′n and

T′

n= μ(ζ(x))Λ′

(Tn(ζ(x))⊗S

), (3.11)

for any nonnegative integer n.

Remark 3.3. As in the AN case, each physical coordinate xi is defined upto an arbitrary translation, which can be fixed in most cases by requiringthat the resulting potentials be invariant under the realization of the Weylgroup of BN type spanned by Πij and Πi.

3.3. CLASSIFICATION OF THE MODELS OF BCN TYPE

We shall now perform a complete classification of the (Q)ES spin Calogero–Sutherland models determined by quadratic combinations of the form (3.3).In contrast to the AN case, the vector spaces 〈J−i , J0i , J+i 〉, i = 1, . . . , N ,are no longer invariant under the full projective action (2.21)–(2.23) ofGL(2, IR). However, these vector spaces are still form-invariant under in-versions and scalings. Indeed, consider the mappings

Jεi (w) $→ Jεi (z) =(∏k

zmk

)Jεi (w(z))

(∏k

z−mk), zj =

1

wj; (3.12a)

Jεi (w) $→ Jεi (z) = Jεi (w(z)), zj = λwj , (3.12b)

where i, j = 1, . . . , N and λ �= 0 is real or purely imaginary.


Lemma 3.4 The transformed Dunkl operators Jεi under the mappings (3.12)are respectively given by

J−i = −J+i∣∣∣b′→b , J0i = −J0i , J+i = −J−i

∣∣∣b→b′ ; (3.13a)

J−i = λJ−i , J0i = J0i , J+i = λ−1J+i . (3.13b)

It follows from the previous lemma that the transformed of the operatorH under the mappings (3.12) is still of the form (3.3), with (in general)different coefficients c++, c00, c−− and c0. Using these transformations, thepolynomial P in (3.6) can be reduced to one of the following canonicalforms:

1) 1 ,

2) ± νz2 ,3) ± ν(1 + z2) ,4) ± ν(1− z2)2 ,

5) ν(e2iθ − z2)(e−2iθ − z2) ,6) ± ν(1− z2)(1 − k2z2) ,7) ν(1− z2)(1 − k2 + k2z2) ,

where ν > 0, 0 < k < 1, and 0 < θ ≤ π/4. It turns out that for eachof these canonical forms except the second ones, the resulting potentialsare invariant under the Weyl group of BN type provided the arbitraryconstants which appear in the change of variables (3.7) are suitably chosen.The potential in each case can be expressed as

V (x) =∑i

U(xi) + Vspin(x) ,

where the first term does not contain the spin operators Sij, Si and thuscan be viewed as the contribution of a scalar external field. Without lostof generality, cf. Eq. (2.27), we shall present the list of potentials for aparticular value of the parameter ν. We shall express the external potentialsin terms of three parameters, α, β, β′, where

α = a(N − 1) + 1

2(b+ b′ +m) , (3.14)

and β, β′ are given in Table II. We shall also drop some inessential constantterms which commute Λ′ and therefore preserve R′n and T′n for all n. Werefer the reader to Ref. [20] for the explicit expressions of the change ofvariable and gauge factor in each case.

Case 1. P (z) = 1 .Scalar external potential :

U(x) = βx2 .


Spin potential :

Vspin(x) = a∑i�=j

[(x−ij)

−2 (a+ Sij) + (x+ij)−2 (a+ Sij)

]+ b

∑i

x−2i (b+ Si) .

Case 2a. P (z) = 4 z2 .Scalar external potential :

U(x) = 0 .

Spin potential :


[sinh−2 x−ij (a+ Sij)− cosh−2 x−ij (a+ Sij)

].

Case 2b. P (z) = −4 z2 .Scalar external potential :

U(x) = 0 .

Spin potential :


[sin−2 x−ij (a+ Sij) + cos

−2 x−ij (a+ Sij)].

Case 3a. P (z) = 4 (1 + z2) .Scalar external potential :

U(x) = −4β(β − 1) cosh−2 2x .

Spin potential :


[( sinh−2 x−ij − cosh−2 x+ij) (a+ Sij)

+ ( sinh−2 x+ij − cosh−2 x−ij) (a+ Sij)]

+ 4b∑i

sinh−2 2xi (b+ Si) .

Case 3b. P (z) = −4 (1 + z2) .Scalar external potential :

U(x) = 4β(β − 1) cos−2 2x .


Spin potential :


[( sin−2 x−ij + cos

−2 x+ij) (a+ Sij)

+ ( sin−2 x+ij + cos−2 x−ij) (a+ Sij)

]+ 4b

∑i

sin−2 2xi (b+ Si) .

Case 4a. P (z) = (1 − z2)2 .Scalar external potential :

U(x) = 2β2 cosh 4x+ 4β(1 + 2α) cosh 2x .

Spin potential :


[sinh−2 x−ij (a+ Sij) + sinh

−2 x+ij (a+ Sij)]

+ b∑i

sinh−2xi (b+ Si)− b′∑i

cosh−2xi (b′ + (−1)mSi).

Case 4b. P (z) = −(1− z2)2 .Scalar external potential :

U(x) = −2β2 cos 4x− 4β(1 + 2α) cos 2x .

Spin potential :


[sin−2 x−ij (a+ Sij) + sin

−2 x+ij (a+ Sij)]

+ b∑i

sin−2xi (b+ Si) + b′∑i

cos−2xi (b′ + (−1)mSi) .

Case 5. P (z) = (e2iθ − z2)(e−2iθ − z2) .The modulus of the elliptic functions in this case is k = cos θ.

Scalar external potential :

U(x) = 4k′2dn−22x[β2 − α(α+ 1)− kβ

k′(1 + 2α) cn 2x

].


Spin potential :


[(dn2x−ijsn2x−ij

− k2k′2sn2x+ij

dn2x+ij

)(a+ Sij)

+

(dn2x+ij

sn2x+ij− k2k′2

sn2x−ijdn2x−ij

)(a+ Sij)

]

+ b∑i

( cnxisnxi dnxi

)2(b+ Si)

+ b′∑i

(snxi dnxicnxi

)2(b′ + (−1)mSi) .

Case 6a. P (z) = 4(1 − z2)(1− k2 z2) .Here and in the remaining cases the elliptic functions have modulus k.

Scalar external potential :

U(x) = 4k′2[β(β − 1) cn−22x− β′(β′ − 1) dn−22x

].

Spin potential :


[(cn2x−ijdn

2x−ijsn2x−ij

+ k′4sn2x+ij

cn2x+ijdn2x+ij

)(a+ Sij)

+

(cn2x+ijdn

2x+ij

sn2x+ij+ k′4

sn2x−ijcn2x−ijdn

2x−ij

)(a+ Sij)

]+ 4b

∑i

sn−22xi (b+ Si) + 4k2b′∑i

sn22xi (b′ + (−1)mSi) .

Case 6b. P (z) = −4(1− z2)(1− k′2 z2) .Scalar external potential :

U(x) = 4[β(β − 1) k2sn22x− β′(β′ − 1) k′2dn−22x

].

Spin potential :


[(dn2x−ij

sn2x−ijcn2x−ij

+ k4sn2x+ijcn

2x+ij

dn2x+ij

)(a+ Sij)

+

(dn2x+ij

sn2x+ijcn2x+ij

+ k4sn2x−ijcn

2x−ijdn2x−ij

)(a+ Sij)

]+ 4b

∑i

sn−22xi (b+ Si) + 4k′2b′∑i

cn−22xi (b′ + (−1)mSi) .


Case 7. P (z) = 4(1− z2)(k′2 + k2 z2) .Scalar external potential :

U(x) = 4[β′(β′ − 1) k2sn22x+ β(β − 1) k′2cn−22x

].

Spin potential :


[(cn2x−ij

sn2x−ijdn2x−ij

+sn2x+ijdn

2x+ij

cn2x+ij

)(a+ Sij)

+

(cn2x+ij

sn2x+ijdn2x+ij

+sn2x−ijdn

2x−ijcn2x−ij

)(a+ Sij)

]+ 4b

∑i

sn−22xi (b+ Si)− 4k′2b′∑i

dn−22xi (b′ + (−1)mSi) .

Remark 3.5. The potentials in Cases 1, 2, and 3 are ES for all valuesof the parameters. The potentials of type 4 are also ES for β = 0. Theelliptic potentials in Cases 5–7 are all QES. The potentials in Case 2 arenot invariant under the BN Weyl group, due to the fact that a sign reversalin the variable zi yields a translation in the corresponding physical variablexi.

Remark 3.6. The potential in Case 1 is the rational BN -type model in-troduced by Yamamoto [68] and later studied by Dunkl [16]. Case 4b forβ = 0 is Yamamoto’s BN -type trigonometric potential with λ1 = −b (inthe notation of Ref. [68]), and either λ′1 = −b′ for m even or λ′1 = b′ form odd. The remaining potentials were introduced in [20]. The spectrumand integrability of the hyperbolic Yamamoto model (Case 4a) have beenrecently studied in detail in Ref. [21].

Remark 3.7. The potentials in the classification diverge as (x−ij)−2 at the

hyperplanes xi = xj, 1 ≤ i < j ≤ N . Depending on the case, the potentialmay possess inverse-square singularities at some other hyperplanes. Thephysical eigenfunctions are required to vanish faster than the square rootdistance to any of these hyperplanes, which yield some constraints on theparameters defining the potential. In addition, when the singular hyper-planes do not divide IRN into bounded disjoint sets, one has to imposefurther constraints in order to ensure the square-integrability of the eigen-functions (see Ref. [20] for the complete list of constraints in each case).For example, in Case 3a the change of variables and gauge factor can berespectively taken as

z = sinh 2x , (3.15)


μ(x) =∏i<j

∣∣∣ sinh 2x−ij sinh 2x+ij ∣∣∣a ∏i

| sinh 2xi|b coshβ 2xi , (3.16)

where β is given in Table II. It follows that the spin functions ψ ∈ R′mvanish at the singularities x±ij = 0 and xi = 0 of the potential faster thanthe square root distance to these hyperplanes and are square-integrableover the domain {x ∈ IRN : x1 > · · · > xN > 0} if and only if a > 1/2,b > 1/2 and β < −(2a(N − 1) + b+m).Remark 3.8. In all cases except the second one, we can obtain physical

wavefunctions of a given parity by a suitable definition of the gauge factor ineach of the domains determined by the singular hyperplanes. For instance,in Case 3a, the wavefunctions in R′m (with the gauge factor (3.16)) are anti-symmetric under simultaneous sign-reversal of any particle’s coordinate andspin. Note that we can also construct odd-parity fermionic wavefunctionsby replacing Λ′ by any of the projectors

Λ

2N

∏i

(1 + Πi) ,Λ

2N

∏i

(1±Πi) ,

where Λ is the symmetrizer under particles’ permutations, and redefin-ing the star mapping (3.5) and the gauge factor appropriately (see Re-mark 2.12).

We shall now present a formula which describes all the BCN potentialsin the classification in a unified way. In the first place, the spin potentialcan be written (barring some irrelevant constant terms) as


[(v(x−ij) + v(x

+ij + P1)

)(a+ Sij)

+(v(x+ij) + v(x

−ij + P1)

)(a+ Sij)

]+ b

∑i

(v(xi) + v(xi + P1))(b+ Si)

+ b′∑i

(v(xi + P2) + v(xi + P1 + P2))(b′ + (−1)mSi) ,

(3.17)where v is a (possibly degenerate) elliptic function with primitive half-periods P1 and P2 (see Table III). In this formula, expressions like v(x+Pi)or v(x+P1+P2) are defined as zero if any of the periods are infinite. Usingthis notation, in Cases 3 and 5–7 the scalar external potential U(x) can bewritten as

U(x) = λ(λ− 1)[v(x+ 1

2P1) + v(x− 12P1)

]+ λ′(λ′ − 1)

[v(x+ 1

2P1 + P2) + v(x− 12P1 + P2)

],

(3.18)


where λ = β, λ′ = 0 in Cases 3, λ = −α+ iβ, λ′ = −α− iβ in Case 5, andλ = β, λ′ = β′ in Cases 6–7. In Cases 1 and 4 Eq. (3.18) cannot be directlyapplied, since in these cases all the terms in (3.18) are either indeterminateor zero. However, the potentials in Cases 4a and 4b can be obtained fromthat of Case 5 in the limits θ → 0 and θ → π/2, respectively. Likewise,applying the rescaling xi $→ νxi (i = 1, . . . , N , ν > 0) to the potential oftype 3a or 3b one obtains the potential of type 1 by replacing β by β/(4ν2)and letting ν → 0.

In Cases 5–7, the function v that determines the potential V (x) can beexpressed up to an irrelevant additive constant in terms of the Weierstrass℘ function as

v(x) = ℘(x;P1, P2) , (3.19)

where the primitive half-periods P1 and P2 are listed in Table III. Sub-stituting Eq. (3.19) into Eqs. (3.17) and (3.18), and applying a modulartransformation of the Weierstrass function [45] to the one-particle termswe obtain the following remarkable formula for the potential V (x) in Cases5–7:

V (x) = a∑i�=j

[(℘(x−ij ;P1, P2) + ℘(x

+ij + P1;P1, P2)

)(a+ Sij)

+(℘(x+ij ;P1, P2) + ℘(x

−ij + P1;P1, P2)

)(a+ Sij)

]+ 4b

∑i

℘(2xi;P1, 2P2)(b+ Si)

+ 4b′∑i

℘(2xi + 2P2;P1, 2P2)(b′ + (−1)mSi)

+ 4∑i

[λ(λ− 1)℘(2xi + P1;P1, 2P2)

+ λ′(λ′ − 1)℘(2xi + P1 + 2P2;P1, 2P2)].

(3.20)One of the main results of Ref. [20] is the fact that the potential (3.20) isQES provided that the ordered pair (P1, P2) is chosen from Cases 5–7 inTable III. In fact, the remaining BCN -type (Q)ES spin potentials in Cases1–4 can be obtained from the potentials in Eq. (3.20) by sending one orboth of the half-periods of the Weierstrass function to infinity. This is ofcourse reminiscent of the analogous formula for integrable scalar Calogero–Sutherland models associated to root systems [48].

4. Conclusions

In this article we have given an overview of a novel systematic methodfor constructing (Q)ES scalar or spin Calogero–Sutherland models using


Dunkl operators. Our approach is close in spirit to the hidden symmetryalgebra approach to scalar (Q)ES models described in Section 1. The quasi-exact solvability of the Hamiltonian is a consequence of the invariance ofthe polynomial space Rm under all three families of Dunkl operators (of agiven type). We emphasize that all the previously known ES spin Calogero–Sutherland models are obtained as particular cases, as well as several newfamilies of (Q)ES spin CS models, some of them of elliptic type. Althoughin this review we have focused on the construction of new (Q)ES spin CSmodels, the spectrum and the integrability of these models can also bestudied using the properties of Dunkl operators.

Another important development not covered in this review is the connec-tion between Calogero–Sutherland models and integrable spin chains withlong-range interactions of Haldane–Shastry type [33, 58]. In fact, every spinCalogero–Sutherland model can be related to a spin chain with long-rangeinteractions by taking the strong coupling limit, in which the particles are“frozen” in their classical equilibrium positions, and their interaction arerestricted to the spin degrees of freedom [53, 63]. In this limit, the constantsof motion of the dynamical CS model yield those of the spin chain, therebyproving its integrability [69, 21]. Moreover, in the exactly solvable cases thespectrum and partition function of the spin chain can be derived in principlefrom those of the corresponding scalar and spin dynamical models [54]. Aninteresting open problem in this respect is the generalization of the aboveresults to the new (Q)ES spin CS models presented in Sections 2 and 3.

The potentials presented in Section 2 are all invariant under the real-ization of the AN Weyl group spanned by the total permutation operatorsΠij . A remarkable feature of the potentials in Cases 2–4 and 7–9 is theiradditional invariance under a change of sign of the spatial coordinate ofany particle. Therefore, although these potentials are not invariant underthe full BN Weyl group spanned by total permutation and sign-reversaloperators Πij and Πi, they are invariant under the restriction of the actionof this group to the spatial coordinates. These models thus occupy anintermediate position between the usual spin CS models of AN type andthe fully BN -invariant spin CS models discussed in Section 3.

We note finally that in the AN case, the rational, trigonometric andhyperbolic models with both V + = 0 and V + = V − (see Eq. (2.26)) areobtained in the classification (in Cases 1–6). In contrast, in the elliptic Cases7–9 the interaction potential is always of the form V + = V −. It remains anopen question to determine whether the purely AN (with V + = 0) ellipticpotentials are also QES; see Refs. [14, 8] for partial results in this direction.


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TableI.CoefficientsA,B,C,DoftheexternalpotentialofANtypeinCases1–9.Hereαε=c

ε 4,withε=±,0,andthe

parameter�isgiveninEq.(2.14).

Case

AB

CD

1α2 +

4α0α+

4(α

2 0+2α−α+)

2(8α−α0+�α

+)

2α2 +

2α0α+

α2 0+α+(3�+2α−)

1 4

( (2α−−�)

2−1)

3α2 +

2α+(α

0+�)

2(α

++α−)(α0−�)

(α++α−)2+(α

0−�)

2−1

4−α2 +

2α+(�−α0)

2(α

++α−)(�+α0)

(α++α−)2+(�+α0)2−1

5α2 +

2α+(α

0+�)

2α−(α

0−�)

α2 −

61 2

( (α +−α−)2−α2 0

)2( α2 −−α2 ++�α

0)

α0(α

−−α+)

2( α 0(

α++α−)+�(α+−α−))

7k2(�

2−1)+k2α0

k′2

( α 0 k′2−2�)

+1

k′4(α

++k2α−)2

2

k′2(α

++k2α−)( �−α0

k′2

)�2−1+α0

k′2

( α 0 k′2+2�)

+1

k′4(α

++α−)2

2

k′2(α

++α−)( �

+α0

k′2

)

8�2−1+α0

k2

( α 0 k2−2�)

+1 k4(α

++α−)2

2 k2(α

++α−)( α 0 k2

−�)

k′2(�

2−1)+k′2α0

k2

( α 0 k2+2�)

+1 k4(α

++k′2α−)2

2 k2(α

++k′2α−)( α 0 k2

+�)

9k′2(1−�2)+k′2α0(2�−α0)

+1 k2

( k′2 α+−k2α−

) 22(�−α0)( k′2

α+−k2α−

) (�+α0)2+(α

++α−)2−1

2(�+α0)(α++α−)


TABLE II. Parameters β and β′ of the scalar external potential of BCN type(none of them appear in Cases 2a-2b). The parameter α is given in Eq. (3.14).

Case β β′

1 − 12c0 –

3a c08−(a(N − 1) + b+ m

2

)–

3b −(c08+ a(N − 1) + b+ m

2

)–

4a1

8

(c0 + 2(b− b′)

)–

4b −18

(c0 + 2(b

′ − b))

–

5 − 18kk′

(c0 + 2(k

2 − k′2)(b− b′))

–

6a − 18k′2

(c0 + 4(1 + k2)(b− b′)

)− α 1

8k′2

(c0 + 4(1 + k2)(b− b′)

)− α

6b 18k2

(c0 + 4(1 + k′2)(b′ − b)

)− α − 1

8k2

(c0 + 4(1 + k′2)(b′ − b)

)− α

7 −(c08+ 1

2(k2 − k′2)(b′ − b) + α

)c08+ 1

2(k2 − k′2)(b′ − b)− α


TABLE III. Function v(x) and its primitive half-periodsPi (see Eq. (3.17)) for each of the BCN -type potentials.The constant K ≡ K(k) is the complete elliptic integral ofthe first kind, K′ = K(k′), k is the modulus of the ellipticfunctions, and k′ =

√1− k2.

Case v(x) P1 P2

1 x−2 ∞ ∞

3a sinh−2 xiπ

2∞

3b sin−2 xπ

2∞

4a sinh−2 x ∞ iπ

2

4b sin−2 x ∞ π

2

5dn2x

sn2xK + iK′ K

6acn2x dn2x

sn2xK

iK′

2

6bdn2x

sn2x cn2xiK′

K

2

7cn2x

sn2x dn2xK

1

2(K + iK′)

YANG–BAXTER MAPS AND MATRIX SOLITONS

V.M. GONCHARENKO ([email protected])Chair of Mathematics and Financial Applications, FinancialAcademy, Leningradsky prospect, 49, Moscow, Russia

A.P. VESELOV ([email protected])Department of Mathematical Sciences, LoughboroughUniversity, Loughborough, Leicestershire, LE 11 3TU, UKLandau Institute for Theoretical Physics, Kosygina 2, Moscow,117940, Russia

Abstract. New examples of the Yang–Baxter maps (or set-theoretical solutions to thequantum Yang–Baxter equation) on the Grassmannians, arising from the theory of thematrix KdV equation are discussed. The Lax pairs for these maps are produced usingthe relations with the inverse scattering problem for the matrix Schrodinger operator

1. Introduction.

The problem of studying the set-theoretical solutions to the quantum Yang–Baxter equation was suggested by V.G. Drinfeld [1]. This stimulated re-search in this direction, mainly from the algebraic point of view (see e.g.[2],[3]). The dynamical aspects of this problem were discussed in the paper[4] where also a shorter term “Yang–Baxter map” for such solutions wassuggested.

In this paper we present some new examples of the Yang–Baxter mapsappeared in relation with the theory of solitons. In the case when thesolitons have internal degrees of freedom described by some manifold Xtheir pairwise interaction gives a map from X×X into itself which satisfiesthe Yang–Baxter relation, which means that the final result of multiparticleinteraction is independent of the order of collisions (see Kulish’s paper [5]which is the first one we know containing such a statement).

As an example of the equation with the soliton solutions having non-trivial internal parameters we consider the matrix KdV equation

Ut = 3UUx + 3UxU − Uxxx , (1.1)

191


192 V.M. GONCHARENKO AND A.P. VESELOV

where U is n × n matrix. This equation was introduced in the famousP. Lax’s paper [6] and was the subject of investigations in several pa-pers including [7–9]. The related inverse scattering problem for the matrixSchrodinger operator was investigated by Martinez Alonso and Olmedilla[10, 11].

We will show that the formulas from [9] for two matrix KdV solitoninteraction can be generalized to determine some Yang–Baxter maps on theGrassmanniansG(k, n) and products of two GrassmanniansG(k, n)×G(n−k, n).We produce also the Lax pairs for these maps using the relations withthe inverse scattering problem for the matrix Schrodinger operator [10, 11].

2. Two-soliton interaction as Yang–Baxter map.

Let us start with the definition of the Yang–Baxter map (cf. [1], [4]). Let Xbe any set and R be a map: X ×X → X×X. Let Rij : Xn → Xn, Xn =X ×X× .....×X be the maps which acts as R on i-th and j-th factors andidentically on the others. If P : X2 → X2 is the permutation: P (x, y) =(y, x), then

R21 = PRP.

The map R is called Yang–Baxter map if it satisfies the Yang–Baxterrelation

R12R13R23 = R23R13R12, (2.1)

considered as the equality of the maps ofX×X×X into itself. If additionallyR satisfies the relation

R21R = Id, (2.2)

we will call it reversible Yang–Baxter map.We will actually consider the parameter-dependent Yang–Baxter maps

R(λ, μ) (λ, μ ∈ C) satisfying the corresponding version of Yang–Baxterrelation

R12(λ1, λ2)R13(λ1, λ3)R23(λ2, λ3) = R23(λ2, λ3)R13(λ1, λ3)R12(λ1, λ2)(2.3)

and reversibility condition

R21(μ, λ)R(λ, μ) = Id. (2.4)

Although this case can be considered as a particular case of the previousone by introducing X = X ×C and R(x, λ; y, μ) = R(λ, μ)(x, y) it is moreconvenient for us to keep the parameter separately.

To construct examples of such maps consider the two-soliton interactionin the matrix KdV equation (1.1). At the beginning let U be a general n×ncomplex matrix, no symmetry conditions are assumed.

YANG–BAXTER MAPS AND MATRIX SOLITONS 193

It is easy to check that the matrix KdV equation has the soliton solutionof the form

U = 2λ2P sech2(λx− 4λ3t),where P must be a projector: P 2 = P. If we assume that P has rank 1 then

P should have the form P = ξ ⊗ η(ξ, η)

. Here ξ is a vector in a complex vector

space V of dimension d, η is a vector from the dual space V ∗ (covector)and bracket (ξ, η) means the canonical pairing between V and V ∗.

To find the two-soliton solutions one can use the inverse scattering prob-lem for the general matrix Schrodinger operator developed in [10, 11]. Thecorresponding formulas have been found and analyzed in [9]. In particular,it was shown that the change of the matrix amplitudes P (“polarizations”)of two solitons with velocities λ1 and λ2 after their interaction is describedby the following map:

R(λ1, λ2) : (ξ1, η1; ξ2, η2)→ (ξ1, η1; ξ2, η2)

ξ1 = ξ1 +2λ2(ξ1, η2)

(λ1 − λ2)(ξ2, η2)ξ2, η1 = η1 +

2λ2(ξ2, η1)

(λ1 − λ2)(ξ2, η2)η2, (2.5)

ξ2 = ξ2 +2λ1(ξ2, η1)

(λ2 − λ1)(ξ1, η1)ξ1, η2 = η2 +

2λ1(ξ1, η2)

(λ2 − λ1)(ξ1, η1)η1. (2.6)

We claim that this map is a reversible parameter-dependent Yang–Baxter map. This can be checked directly although the calculations arequite long.

A better way is explained in the next section.

3. Matrix factorizations and Lax pairs.

Suppose we have a matrix A(x, λ; ζ) depending on the point x ∈ X, aparameter λ and an additional parameter ζ ∈ C, which we will call spectralparameter. We assume that A depends on ζ polynomially or rationally. Thecase of elliptic dependence is also very interesting (see [13]) but we will notconsider it here.

Consider the product L = A(y, μ; ζ)A(x, λ; ζ), then change the orderof the factors L → L = A(x, λ; ζ)A(y, μ; ζ) and re-factorize it as: L =A(y, μ; ζ)A(x, λ; ζ). Suppose that this re-factorization relation

A(x, λ; ζ)A(y, μ; ζ) = A(y, μ; ζ)A(x, λ; ζ) (3.1)

uniquely determines x, y.


It is easy to see that the map

R(λ, μ)(x, y) = (x, y) (3.2)

determined by (3.1) satisfies the Yang–Baxter relation. Indeed if we considerthe product A(x1)A(x2)A(x3) (we omit here the parameters λi and ζ forshortness) then applying the left hand side of (2.1) to this product we have

A(x1)A(x2)A(x3) = A(x(1)1 )A(x

(1)3 )A(x

(1)2 )

= A(x(2)3 )A(x

(2)1 )A(x

(2)2 )

= A(x(3)3 )A(x

(3)2 )A(x

(3)1 ).

Similarly the right hand side corresponds to the relations

A(x1)A(x2)A(x3) = A(x(1)2 )A(x

(1)1 )A(x

(1)3 )

= A(x(2)2 )A(x

(2)3 )A(x

(2)1 )

= A(x(3)3 )A(x

(3)2 )A(x

(3)1 ).

If the factorization is unique we have x(3)i = x

(3)i , which is exactly the

Yang–Baxter relation.If a parameter-dependent Yang–Baxter map R(λ, μ) can be described in

such a way we will say that A(x, λ; ζ) is a Lax pair for R. As it was shownin [4] such a Lax pair allows to produce the integrals for the dynamics ofthe related transfer-maps.

Let us come back now to matrix solitons. We claim that the map de-scribed by the formulas (2.5),(2.6) has the Lax pair of the following form(Yuri Suris suggested a simple explanation of this form which works also fora wide class of the Yang–Baxter maps (see [15])) motivated by the inversespectral problem for the matrix Schrodinger operator [11]:

A(ξ, η, λ; ζ) = I +2λ

ζ − λξ ⊗ η(ξ, η)

(3.3)

In the soliton theory this type of matrices were first used by Zakharov andShabat [14].

One can check directly that re-factorization relation for this matrixleads to the map (2.5, 2.6) but we would prefer to do this in a more generalsituation.


4. Generalization: Yang–Baxter maps on the Grassmannians.

Let V be an n-dimensional real (or complex) vector space, P : V → V bea projector of rank k: P 2 = P . Any such projector is uniquely determinedby its kernel K = KerP and image L = ImP, which are two subspaces ofV of dimensions k and n − k complementary to each other: K ⊕ L = V.The space of all projectors X of rank k is an open set in the product of twoGrassmannians G(k, n)×G(n − k, n).

Consider the following matrix

A(P, λ; ζ) = I +2λ

ζ − λP (4.1)

and the related re-factorization relation(I +

2λ1ζ − λ1

P1)(I +

2λ2ζ − λ2

P2)=

(I +

2λ2ζ − λ2

P2)(I +

2λ1ζ − λ1

P1)(4.2)

which we can rewrite in the polynomial form as

((ζ − λ1)I + 2λ1P1)((ζ − λ2)I + 2λ2P2)= ((ζ − λ2)I + 2λ2P2)((ζ − λ1)I + 2λ1P1). (4.3)

We claim that if λ1 �= ±λ2 it has a unique solution. This follows fromthe general theory of matrix polynomials (see e.g. [12]) but in this case wecan see this directly.

Indeed let us compare the kernels of both sides of the relation (4.3)when the spectral parameter ζ = λ1. In the right hand side we obviouslyhave K1 while the left hand side gives

((λ1 − λ2)I + 2λ2P2)−1K1 =(I +

2λ2λ1 − λ2

P2)−1

K1.

Now we use the following property of the matrix (4.1):

A(P,−λ; ζ) = A(P, λ; ζ)−1 (4.4)

to have

K1 =(I − 2λ2

λ1 + λ2P2

)K1. (4.5)

Similarly taking the image of both sides of (4.3) at ζ = λ2 we will have

L2 =(I +

2λ1λ2 − λ1

P1)L2. (4.6)


To find K2 and L1 one should take first the inverse of both sides of(4.2), use the property (4.4) and then repeat the procedure. This will leadus to the formulas:

K2 =(I − 2λ1

λ1 + λ2P1

)K2 (4.7)

and

L1 =(I +

2λ2λ1 − λ2

P2)L1. (4.8)

The formulas (4.5,4.6,4.7,4.8) determine a parameter-dependent Yang–Baxter map on the set of projectors. One can easily check that for k = 1one has the formulas (2.5, 2.6) for two matrix soliton interaction.

If we supply now our vector space V with the Euclidean (Hermitian)structure and consider the self-adjoint projectors P of rank k then thecorresponding space X will coincide with the Grassmannian G(k, n) such aprojector is completely determined by its image L (which is a k-dimensionalsubspace in V and thus a point in G(k, n)) since the kernel K in this caseis the orthogonal complement to L.

The corresponding Yang–Baxter map R on the Grassmannian is deter-mined by the formulas

L1 =(I +

2λ2λ1 − λ2

P2)L1, (4.9)

L2 =(I +

2λ1λ2 − λ1

P1)L2. (4.10)

It would be very interesting to investigate the dynamics of the corre-sponding transfer-maps [4]. As we have shown here the Lax pair for themis given by (4.1).

Acknowledgements

The second author (A.P.V.) is grateful to the organizers and participants ofthe NEEDS conference and NATO Advanced Research Workshop on “NewTrends in Integrability and Partial Solvability” (Cadiz, 10-15 June 2002)and SIDE-V conference (Giens, 21-26 June 2002) where these results werefirst presented and especially to P.P. Kulish, A.B. Shabat and Yu. Suris forstimulating and helpful discussions.

References

1. V.G. Drinfeld, On some unsolved problems in quantum group theory. In “Quantumgroups” (Leningrad, 1990), Lecture Notes in Math., 1510, Springer, 1992, p. 1–8.


2. A. Weinstein, P. Xu, Classical solutions of the quantum Yang–Baxter equation.Comm. Math. Phys. 148, 309–343 (1992).

3. P. Etingof, T. Schedler, A. Soloviev, Set-theoretical solutions to the quantum Yang–Baxter equation. Duke Math. J. 100, (1999).

4. A.P. Veselov, Yang–Baxter maps and integrable dynamics. math.QA/0205335.5. P.P. Kulish Factorization of the classical and quantum S-matrix and conservation

laws. Theor. Math. Phys. 26, 132–137 (1976).6. P.D. Lax., Integrals of nonlinear equations of evolution and solitary waves. Comm.

Pure Appl. Math. 21, 467–490 (1968).7. M. Wadati, T. Kamijo, On the extension of inverse scattering method. Prog. Theor.

Physics 52, 397–414 (1974).8. F. Calogero, A. Degasperis, Nonlinear evolution equations solvable by the inverse

spectral transform. II. Nuovo Cimento. 39B, 1 (1977).9. V.M. Goncharenko,Multisoliton solutions of the matrix KdV equation. Theor. Math.

Phys. 126, 81–91 (2001).10. L. Martinez Alonso, E. Olmedilla, Trace identities in the inverse scattering trans-

form method associated with matrix Schrdinger operators. J. Math. Phys. 23, 2116(1982).

11. E. Olmedilla, Inverse scattering transform for general matrix Schrdinger operatorsand the related symplectic structure. Inverse Problems 1, 219–236 (1985).

12. I. Gohberg, P. Lancaster, L. Rodman, Matrix polynomials. New York: AcademicPress, 1982.

13. A. Odesskii, Set-theoretical solutions to the Yang–Baxter relation from factorizationof matrix polynomials and theta-functions. math.QA/0205051.

14. V.E. Zakharov, A.B.Shabat, Integration of the nonlinear equations of mathematicalphysics by the method of the inverse scattering problem. II. Funct. Anal. Appl. 13,13–22 (1979).

15. Yu.B. Suris, A.P. Veselov Lax pairs for Yang–Baxter maps. In preparation.

NONLOCAL SYMMETRIES AND GHOSTS

PETER J.OLVER (olver.math.umn.edu)∗Department of Mathematics, University of Minnesota,Minneapolis, MN, USA 55455

1. Introduction

The local theory of symmetries of differential equations has been well-established since the days of Sophus Lie. Generalized, or higher ordersymmetries can be traced back to the original paper of Noether, [32],but were not exploited until the discovery that they play a critical rolein integrable (soliton) partial differential equations, cf. [30, 33, 35].

While the local theory is very well developed, the theory of nonlocalsymmetries of nonlocal differential equations remains incomplete. Particu-lar results on certain classes of nonlocal symmetries and nonlocal differentialequations have been developed by several groups, including Abraham–Shrauner et. al., [1–3, 13], Bluman et. al., [5–7], Chen et. al., [8–10], Fushchichet. al., [17], Guthrie and Hickman, [20–22], Ibragimov et. al., [4], [23, Chap-ter 7], and many others, [11, 12, 16, 18, 19, 24, 28, 29, 31, 37]. Perhaps themost promising proposed foundation for a general theory of nonlocal sym-metries is the Krasilshchik-Vinogradov theory of coverings, [25–27, 38, 39].However, their construction relies on the a priori specification of the un-derlying differential equation, and so, unlike local jet space, does not forma universally valid foundation for the theory.

One of the reasons for the lack of a proper foundation is a continuinglack of understanding of the calculus of nonlocal vector fields. Recently, [34],during an attempt to systematically investigate the symmetry propertiesof the Kadomtsev–Petviashvili (KP) equation, Sanders and Wang madea surprising discovery that the Jacobi identity for nonlocal vector fieldsappears to fail! The observed violation of the naıve version of the Jacobi

∗ Research supported in part by NSF Grant DMS 98–03154.

199


200 P.J. OLVER

identity applies to all of the preceding nonlocal symmetry calculi, and, con-sequently, many statements about the “Lie algebra” of nonlocal symmetriesof differential equations are, by and large, not valid as stated. This indicatesthe need for a comprehensive re-evaluation of all earlier results on nonlocalsymmetry algebras.

In this paper, I propose a new theoretical and computational basis fora nonlocal theory which, like the original jet bundle construction, doesnot rely on a specific differential equation, but applies equally well to awide variety of nonlocal systems. I will also review the concept of a ghostsymmetry, introduced in [34], that resolves the apparent Jacobi paradox.Applications to the classification of symmetries of the KP equation appearin [34]. Similar issues appear in the study of recursion operators by Sandersand Wang, [36].

2. Generalized symmetries

Let us recall the basic theory of generalized symmetries in the local jetbundle framework as presented in [33]. We specify p independent variablesx = (x1, . . . , xp) and q dependent variables u = (u1, . . . , uq). The inducedjet space coordinates are denoted by

uαJ =∂#Juα

(∂x1)j1 · · · (∂xp)jp ,

in which 1 ≤ α ≤ q, and J = (j1, . . . , jp) ∈ INp is a (nonnegative) multi-index, so jν ≥ 0, of order #J = j1 + · · · + jp. We let u(∞) = (. . . uαJ . . .)denote the collection of all such local jet variables. A differential functionis a smooth function P [u] = P (x, u(∞)) depending on finitely many jetvariables. If u = f(x) is any smooth function, we let P [f ] denote theevaluation of the differential function P on f .

The total derivatives D1, . . . ,Dp are defined so that DiP [f ] = ∂i(P [f ])where ∂i = ∂/∂x

i. They act on the space of differential functions as deriva-tions, and so are completely determined by their action

Di(xj) = δji , Di(u

αJ ) = u

αJ+ei , (2.1)

on the coordinate functions. Here ei ∈ INp denotes the ith basis multi-indexhaving a 1 in the ith position and zeros elsewhere. If J is a multi-index, we

let DJ = Dj11 · · ·Djpp denote the corresponding higher order total derivative;

in particular, uαJ = DJuα.

We consider generalized vector fields in evolutionary form

v = vQ =q∑α=1

∑J≥0

DJQα∂

∂uαJ, (2.2)

NONLOCAL SYMMETRIES AND GHOSTS 201

where Q = (Q1, . . . , Qq) is the characteristic, and serves to uniquely specifyv. We note the basic formula

vQ(P ) = DP (Q) (2.3)

where DP denotes the Frechet derivative of the differential function P , [33],which is a total differential operator with components

DαP =∑J

∂P

∂uαJDJ , α = 1, . . . , q. (2.4)

The Lie bracket or commutator between two evolutionary vector fields isagain an evolutionary vector field

[vP ,vQ] = v[P,Q],

with characteristic

[P,Q] = vP (Q)− vQ(P ) = DQ(P )−DP (Q). (2.5)

The Lie bracket satisfies the Jacobi identity, and hence endows the spaceof evolutionary vector fields with the structure of a Lie algebra.

3. Counterexamples to the Jacobi identity?

Attempting to generalize the algebra of evolutionary vector fields to nonlo-cal variables runs into some immediate, unexpected difficulties. Intuitively,the nonlocal variables should be given by iterating the inverse total deriva-tives D−1i , applied to either the jet coordinates uαJ , or, more generally, todifferential functions. In particular, we allow nonlocal variables uαJ = D

Juα

in which J ∈ ZZp is an arbitrary multi-index. Even more generally, onemight allow inversion of arbitrary total differential operators D−1, whereD =∑

K PK [u]DK , whose coefficients PK can be either constants, or even

general differential functions.However, the following fairly simple computation appears to indicate

that the Jacobi identity does not hold between nonlocal vector fields.

Example 3.1. Let p = q = 1, with independent variable x and dependentvariable u. Consider the vector fields v,w, and z with respective charac-teristics 1, ux and D

−1x u. The first two are local vector fields, and, in fact,

correspond to the infinitesimal generators of the translation group

(x, u) $−→ (x+ δ, u + ε).

The Jacobi identity for these three vector fields has the form

[1, [ux,D−1x u]] + [ux, [D

−1x u, 1]] + [D

−1x u, [1, ux]] = 0, (3.1)

202 P.J. OLVER

where we work on the level of the characteristics, using the induced com-mutator bracket (2.5). Since

[1, ux] = Dux(1) −D1(ux) = Dx(1) = 0, (3.2)

reflecting the fact that the group of translations is abelian, we only needto compute the first two terms in (3.1). First, using the definition of theFrechet derivative, we compute

[ux,D−1x u] = DD−1x u(ux)−Dux(D−1x u)

= D−1x ux −Dx(D−1x u) = u+ c− u = c,

where c is an arbitrary constant representing the ambiguity in the an-tiderivative D−1x . Thus,

[1, [ux,D−1x u]] = [1, c] = 0,

irrespective of the integration constant c. On the other hand,

[D−1x u, 1] = −D−1x (1) = −x+ d,

where d is another arbitrary constant, and so

[ux, [D−1x u, 1]] = [ux,−x+ d] = −Dx(−x+ d) = 1.

Therefore, no matter how we choose the integration “constants” c, d, theleft hand side of (3.1) equals 1, not zero, and so the Jacobi identity appearsto be invalid!

This example is, in fact, the simplest of a wide variety of apparentnonlocal counterexamples to the Jacobi identity. Similar problems arise inthe structure of the Lie algebra of nonlocal symmetries of the KP equation,[34], and the theory of recursion operators, [36].

4. Nonlocal differential algebra

In order to keep the constructions reasonably simple, we will work entirelywithin the polynomial category throughout. Thus, we only consider dif-ferential polynomials with polynomial coefficients. Also we work, withoutany significant loss of generality, with real-valued polynomials, the complexversion being an easy adaptation.

By a multi-index we mean a p-tuple J = (j1, . . . , jp) ∈ ZZp with integerentries. The order of J is #J = j1 + · · · + jp. The multi-index is positive,written J ≥ 0, if all its entries are positive: jν ≥ 0. We impose a partialorder on the space of multi-indices with J ≤ K if and only if K−J ≥ 0. We


will also impose a total ordering J ≺ K on the multi-indices that respectsdegree, so if #J < #K, then J ≺ K. In particular, degree lexicographicordering is a convenient choice of total order, [14].

Let A = IR[x] = IR{x1, . . . , xp} denote the algebra of polynomial func-tions f(x) depending upon p variables. The derivatives ∂1, . . . , ∂p make Ainto a (partial) differential algebra. Given a possibly infinite set of depen-dent variables U = {. . . uα . . .}, let A{U} denote the differential algebraconsisting of all polynomials in their local derivatives uαJ = DJuα, J ≥0, whose coefficients are polynomial functions in A. We write P [u] =P (. . . xi . . . uαJ . . .) for a differential polynomial in A{U}. Even thoughU may contain infinitely many variables, any differential polynomial P ∈A{U} has only finitely many summands and hence depends on only finitelymany variables uαJ . The set of polynomials that only depend on x can beidentified with A itself, and there is a natural decomposition

A{U} = A⊕A�{U},

where A�{U} consists of all differential polynomials that vanish wheneverwe set all uαJ = 0. Any ordering uα ≺ uβ of U induces an ordering of

the derivatives, so uαJ ≺ uβK whenever uα ≺ uβ or α = β and J ≺ K. This

ordering in turn induces the degree lexicographic ordering on the differentialmonomials in A�{U}.

The total derivatives D1, . . . ,Dp act on A{U} as derivations. Theirkernels are well-known:

Lemma 4.1 The kernel of the ith total derivative is

kerDi = Ai = {f(x1, . . . , xi−1, xi+1, . . . , xp) | f ∈ A}. (4.1)

In particular, the restriction of Di : A�{U} → A�{U} has trivial kernel.

We begin our construction with the algebra of local differential polyno-mials

Bloc = B(0) = A⊕ B(0)� = A{u1, . . . , uq}Our goal is to construct a nonlocal differential algebra

B(∞) = A⊕ B(∞)� ⊃ Bloc

such that each total derivative Di : B∞� −→ B∞∗ defines an invertible mapeverywhere except on the ordinary polynomials f(x) ∈ A. The polynomialsin B∞ will, therefore, be polynomials involving expressions of the formD−1i P where P is any local or nonlocal differential polynomial, e.g. D−1i u,D−1i (u2uj), or even D

−1i (uijD

−1j (u2)D−1k (u2i )), and so on. Our construction

will accomplished by inductively implementing the following construction.

204 P.J. OLVER

At each step, we are given an infinite1 collection of dependent variables

U (m) = U0 ∪ U1 ∪ U2 ∪ · · · ∪ Um,

which is the disjoint union of the subsets

Uk = {uα | depthuα = k}.

consisting of all dependent variables of a given depth. Roughly speaking,the depth of a variable will measure its “depth of nonlocality”. In par-ticular, all the original variables in our local differential algebra U (0) =U0 = {u1, . . . , uq} have depth 0. We also assign a weight wα = wtuα

to each variable in U (m). For simplicity, the original dependent variablesuα ∈ U0 can have weight 1, although the initial weighting can be adaptedto particular applications, as in [35].

We let B(m) = A⊕ B(m)� = A{U (m)} denote the algebra of polynomials

in the variables uαJ = DJuα for all uα ∈ U (m) and J ∈ ZZp. We define

depthuαJ = depthuα, wtuαJ = wtuα.

Note that linearly nonlocal variables uαJ for uα ∈ U0 will continue to have

depth 0. The total derivatives act on B(m) as derivations subject to thesame rules (2.1).

Let M(m) denote the set of x-independent monomials in B(m)� , i.e.,

products of the form M = uα1J1 · · · uαkJk. Therefore, B(m) = A[M(m)] consists

of finite linear combinations of monomials with coefficients in A. We extendthe notion of depth and weight to monomials inM(m) by setting

depth (M N) = max{depthM,depthN},wt (M N) = wtM +wtN, (4.2)

whenever M,N ∈M(m). Thus, we write

M(m) =M0 ∪M1 ∪ · · · ∪Mm,

whereMk denotes the set of monomials of depth k.We describe the induction step. For each monomial Mγ ∈ Mm, and

each 1 ≤ i ≤ p, we introduce a new dependent variable uγ,i ∈ Um+1 ofdepth m+ 1 such that

Diuγ,i =Mγ . (4.3)

Formally, we can writeuγ,i = D−1i Mγ , (4.4)

1 Except the initial step, where we start with only finitely many dependent variablesu1, . . . , uq .


but it is better, for the time being, to regard each of these as a completelynew dependent variable. Once the induction procedure is complete, we shallimpose the relations implied by (4.3). The weight and depth of each newvariable is

wtuγ,i = wtMγ , depthuγ,i = m+ 1 > depthMγ .

The inductive step sets

Um+1 = {uγ,i}, U (m+1) = U (m) ∪ Um+1, B(m+1) = A{U (m+1)}.Finally, we let

U (∞) =∞⋃m=0

Um, B(∞) = A⊕ B(∞)� = A{U (∞)}. (4.5)

We can identify B(∞) as the injective limit B(0) ↪→ B(1) ↪→ B(2) ↪→ · · · ofsubalgebras of progressively higher and higher depth. Note that D−1i P ∈B(∞)� is well-defined for any P ∈ B(∞)

� . A nonlocal differential polynomialis said to be homogeneous of weight k if all its constituent monomials

have weight k. We write B(∞)k for the set of all homogeneous differential

polynomials of weight k, and so B(∞) = A⊕⊕∞k=1 B

(∞)k .

Of course, the differential algebra B(∞) contains a huge number of redun-dancies, since we have not yet taken into account the defining relations (4.3)of our nonlocal variables. Thus, we need to determine which of these nonlo-cal expressions are trivial, meaning that they vanish when evaluated uponany smooth function. In local differential algebra, one can prove triviality byevaluating the differential polynomial on all polynomial functions u = p(x).In the nonlocal case, the class of polynomial functions is not appropriatebecause the inverse derivatives ∂−1i include a possible integration constant,and so are not uniquely defined on the space of polynomial functions. Tocheck the vanishing of a nonlocal differential polynomial, one needs to keeptrack of a consistent choice of integration constants used to evaluate thenonlocal terms, and this rapidly becomes a difficult, if not intractable issue.A more enlightened approach is to introduce the following functions.

Definition 1. A function of the form f(x) = p(x) en·x in which p(x) isa polynomial and n = {n1, . . . , np} ∈ ZZp will be called a polynomial–exponential function. It will be called positive if n > 0, meaning ni > 0 fori = 1, . . . , p. Let F = {p(x) en·x | n > 0} denote the algebra of all positivepolynomial–exponential functions.

The key property is that, in contrast to the space of polynomials, deriva-tives are invertible when restricted to the positive polynomial–exponentialspace F .

206 P.J. OLVER

Lemma 4.2 The derivative ∂i :F → F is a one-to-one linear map, andhence its inverse ∂−1i :F → F is uniquely defined. Consequently, if P ∈B(∞)k is any homogeneous nonlocal differential polynomial, its evaluation onf ∈ F gives a uniquely defined polynomial–exponential function P [f ] ∈ F .Moreover, evaluation commutes with (anti-) differentiation, so DJP [f ] =∂J(P [f ]) for any J ∈ ZZp.

Proof : We recall the well-known formula

D−1i (P Q) =∑j≥0(−1)j (DjiP ) (D

−j−1i Q). (4.6)

If P = p(x) is a polynomial, then the sum terminates, and gives an explicitformula for D−1i (p(x) en·x). Q.E.D.

Definition 2. A homogeneous nonlocal differential polynomial P ∈ B(∞)k

is trivial P [f ] = 0 for all f = (f1, . . . , f q) ∈ Fq.

The fact that testing a nonlocal differential polynomial on all polynomi-al–exponentials is sufficient to detect triviality is a consequence of the factthat polynomial–exponential functions are sufficiently extensive to matchany finite nonlocal jet. We let

I(∞) =∞⊕k=1

I(∞)k , (4.7)

whereI(∞)k = {P ∈ B∞� | P [f ] = 0, for all f ∈ F},

denote the ideal of all trivial nonlocal differential polynomials. If P ∈ I(∞)

then, by the last remark in Lemma 4.2, DJP ∈ I(∞) for any J ∈ ZZp,and hence I(∞) ⊂ B(∞)

� is a homogeneous nonlocal differential ideal. Inparticular, the defining relations (4.3) of our nonlocal variables belong tothe ideal, meaning Diu

γ,i −Mγ ∈ I(∞).Finally, we define our nonlocal differential algebra to be the quotient

algebra

π:B(∞) −→ Q(∞) = A⊕Q(∞)� = B(∞) /I(∞). (4.8)

This algebra incorporates all the relations implied by (4.3) and their (anti-)derivatives. We easily check that D−1i is uniquely defined on all of Q(∞),

and moreover forms an inverse to Di when restricted to Q(∞)� .

Theorem 4.3 If P ∈ Q(∞)� and 1 ≤ i ≤ p, then there exists a unique

Si ∈ Q(∞)� such that P = DiSi. We write Si = D

−1i P .


In practical applications, the key issue is whether we can perform ef-fective computations in the nonlocal differential algebra Q(∞). The mainquestion is how to recognize whether a given differential polynomial P ∈B(∞) lies in the differential ideal I(∞). We assume, without loss of gener-

ality, that P ∈ I(∞)k is homogeneous. Roughly speaking, differentiating the

polynomial P sufficiently often will eventually (and in a finite number ofsteps) produce a purely local differential polynomial P � ∈ Bloc with theproperty that P ∈ I(∞) if and only if P � ∈ I(∞). However, the latter willoccur if and only if P � = 0 in Bloc, which is trivial to check.

In order to implement an algorithm, we extend our original term order-ing to include all the nonlocal variables U (∞). We set uα,i ≺ uβ,j if and onlyif depthuα,i < depthuβ,j, or if depthuα,i = depthuβ,j = m ≥ 1 and thecorresponding monomials satisfy Mα ≺ Mβ in the induced term ordering

onM(m).Although the full differential algebra B(∞) contains a gigantic number

of different variables, any given polynomial

P (. . . xi . . . uαJ . . . uγ,iK . . .) ∈ B(∞)

k

only depends on finitely many of them, and so all computations are finitein extent. Let uγ,iK = DKu

γ,i be the highest order variable occurring in P .

We can assume ki > 0, since otherwise we replace uγ,iK $−→ DK−eiMγ , inaccordance with (4.3), which has smaller depth and hence appears earlierin the term ordering. We write out

P = Pn (uγ,iK )

n +n−1∑�=0

P� (uγ,iK )

�, (4.9)

where each coefficient P� ∈ B(∞) depends on lower order variables uα,lJ ≺uγ,iK and we assume Pn �∈ I(∞). Since Pn is of lower order than P , the lattercondition can be checked by the same algorithm. The derivative

DiP = DiPn (uγ,iK )

n +n−1∑�=0

[DiP� + (�+ 1)P�+1DKMγ ](uγ,iK )

�. (4.10)

does not appear earlier in the term ordering. However, the combination

Pn DiP − P DiPn = Q =n−1∑�=0

Q�(uγ,iK )

� (4.11)

is of lower order than P in uγ,iK . The induction step claims that P ∈ I(∞)

if and only if Q ∈ I(∞), and hence the same algorithm can be used on Q.

208 P.J. OLVER

Since Q is of lower order, we use the same algorithm on Q, and so eventually— but in a finite number of steps — reducing to a purely local differentialpolynomial, as desired.

To prove the claim, equation (4.11) implies that, for any f ∈ F ,Pn[f ] ∂iP [f ]− P [f ] ∂iPn[f ] = Q[f ] = 0.

Therefore,∂

∂xi

(P [f ]

Pn[f ]

)= 0 ,

and hence P [f ] = 0 or Pn[f ] = 0. Now, if Pn �∈ I(∞) is nontrivial, then thejets of polynomial–exponential functions that solve the nonlocal differentialequation Pn[f ] = 0 forms a proper subvariety, and P [f ] = 0 everywhereoutside this subvariety, which, by continuity, implies P [f ] = 0 for all f ∈ F ,and proves the claim. Of course, the implementation of this algorithm mightbe quite lengthy, and so developing more efficient algorithms would be aninteresting research topic.

5. Evolutionary vector fields and symmetries

In this section we extend the space of evolutionary vector fields to ournonlocal differential algebra. Since they are defined as commutators, theJacobi Identity will be automatically valid.

Definition 3. A evolutionary vector field v on a differential algebra B =A ⊕ B(0) is a derivation v:B → B, with A ⊂ kerv, while [v,Di] = 0commutes with all total derivatives.

Remark : If we drop the hypothesis A ⊂ kerv then the only addi-tional derivations that commute with the total derivatives are the partialderivatives ∂/∂xj ; see [33].

Therefore, an evolutionary vector field v must satisfy

v(P +Q) = v(P ) + v(Q), v(xi) = 0 (5.1)

v(P Q) = v(P )Q+ P v(Q), [v,Di] = 0,

for all P,Q ∈ B and i = 1, . . . , p. Each evolutionary vector field is uniquelyspecified by its action v(uαJ ) on the coordinate variables. We denote thespace of evolutionary vector fields by V = V(B). The commutator bracket

[v,w](P ) = v(w(P )) −w(v(P )), P ∈ B,between two evolutionary vector fields endows V with the structure of aLie algebra, satisfying the usual skew symmetry and Jacobi identities. Theproof of the latter is elementary.


Warning : The space of evolutionary vector fields is not a B module.The product P v of P ∈ B and v ∈ V does not commute with totaldifferentiation.

Given an evolutionary vector field v, we define its characteristic Q ∈ Bqto have components

v(uα) = Qα, α = 1, . . . , q.

The commutation condition implies

v(uαJ ) = v(DJuα) = DJv(u

α) = DJQα

for all positive multi-indices J ≥ 0. Thus, in the local situation, an evo-lutionary vector field is uniquely determined by its characteristic. Thisbasic fact is not true in nonlocal differential algebras — there are nonzeroevolutionary vector fields with zero characteristic — and this observationmotivates the following key definition.

Definition 4. An evolutionary vector field γ is called a aK-ghost for someK ∈ ZZp if γ(uαL) = 0 for all L ≥ K and α = 1, . . . , q.

There are no ghost vector fields in a local differential algebra Blocbecause each evolutionary vector field is uniquely determined by its char-acteristic Q. There are, however, positive ghost vector fields; for examplethe vector field with characteristic Q = 1 is a K-ghost for any positivemulti-index K > 0.

Example 5.1. Let us see how the existence of ghost vector fields servesto resolve the Jacobi identity paradox in (3.1). Surprisingly, the problem isnot with the nonlocal vector field z with characteristic D−1x u, but ratherthe local commutator [v,w] corresponding to the vector fields with char-acteristics 1 and ux, respectively. While [v,w] = 0 on the local differentialalgebra Bloc, it is, in fact, a ghost vector field on a nonlocal differentialalgebra. Thus, surprisingly, in a nonlocal setting, the group of translationsis not abelian!

The action of the vector fields on the local variables does not uniquelyspecify their action on the nonlocal variables, due to the presence of possibleintegration constants. However, as we have seen, the integration constantsdo not play a significant role in the resolution of the Jacobi identity paradox,and so we shall fix all the integration constants to be zero by default.Therefore,

v(uk) = Dkx(1) = χk(x) ≡

⎧⎨⎩0, k > 0,x−k

(−k)! , k ≤ 0,

w(uk) = Dx(uk) = uk+1. (5.2)

210 P.J. OLVER

Since v(uk) only depends on x, we have w(v(uk)) = 0, and so

[v,w](uk) = v(uk+1) = χk+1(x).

Therefore, [v,w] = γ is a ghost vector field that satisfies

γ(uk) = χk+1(x) =

⎧⎨⎩0 k ≥ 0,x−k−1

(−k − 1)! k < 0.

This ghost provides the missing term in the Jacobi identity (3.1). Indeed,

[z,γ](u) = −γ(z(u)) = −γ(D−1x u) = −1.

In [34], we introduced a “ghost calculus” for general nonlocal evolution-ary vector fields. The first remark is that only evolutionary vector fieldsthat depend on the independent variables can be ghosts. Indeed, if γ is aK-ghost, then

γ(uJ) = DJ−Kγ(uK) = 0, J ≥ K.

Therefore, if γ(uI) = PI and J ≥ 0 is any positive multi-index such thatJ + I ≥ K, then

0 = γ(uJ+I) = DJγ(uI) = D

JPI ,

and we know that kerDJ ⊂ A, so that PI is a function of x only.

Lemma 5.2 An evolutionary vector field γ is a K-ghost for some K ∈ ZZpif and only if γ(uαJ) = p

αJ (x) is a polynomial function of x1, . . . , xp.

Definition 5. Given a multi-index K ∈ ZZp, define

χK = DK(1) =

⎧⎨⎩ x−K

(−K)! , K ≤ 0,0 otherwise,

where (−K)! =p∏ν=1

(−kν)! .

(5.3)

Definition 6. Given a multi-index J ∈ ZZp, define the basis ghost vectorfield γJ so that γJ(uK) = χJ+K , which is a K-ghost for any K + J > 0.

Proposition 5.3 Every ghost vector field is a linear combination of thebasis ghosts, γ =

∑J cJγJ , where the cJ ∈ IR are constants.


The summation in Proposition 5.3 can be infinite. However, only certain“configurations” of the nonzero coefficients cJ are allowed in order that γmap Q(∞) to Q(∞).

Let us formulate the results in the one dependent variable case whereu ∈ IR, and so q = 1. (The multi-variable case can be found in [34].) Let ussplit the space of evolutionary vector fields V = Vx ⊕ V� where Vx denotesthe space of purely x-dependent vector fields, so v(uK) = pK(x) ∈ A. In thepolynomial category, every v ∈ Vx is a ghost vector field. The remainder, V�,consists of u-dependent vector fields, where v(uK) = D

KQ ∈ B(∞)� . Since

kerDK = {0} on B(∞)� , the evolutionary vector fields in V� are uniquely

determined by their characteristics Q = v(u), and we write v = vQ asin the local category. Thus, to re-emphasize: only the x-dependent vectorfields can be ghosts and hence cause any difficulty in the non-local category.

Corollary 5.4 If B(∞) is a polynomial differential algebra, then any evo-lutionary vector field v ∈ V can be written a linear combination of basisghosts and a u-dependent vector field vQ ∈ V�:

v = vQ +∑J

cJ γJ , whereby v(uK) = DKQ+

∑J

cJ χK+J . (5.4)

To implement a calculus of evolutionary vector fields, we identify avector field with its “characteristic”. The characteristic of the evolutionaryvector field vQ is, as usual, Q. The characteristic of the ghost vector fieldγJ will be formally written as χJ . In this manner, every nonlocal vectorfield (5.4) has a unique characteristic

S = Q+∑J

cJ χJ . (5.5)

In particular, a local vector field with polynomial characteristic xK becomesa ghost characteristic K!χ−K . Indeed, one can, again in the one dependentvariable case, replace all polynomials xK $−→ K!χ−K wherever they appearin the characteristic (5.5). The only place true ghosts appear, i.e., χJ withJ �≤ 0, is in the u-independent terms in the summation. Only when thevector field has been evaluated on a nonlocal differential polynomial are weallowed to replace the ghost functions χJ by their actual formulas (5.3).

In this calculus, the product rule xJ xK = xJ+K becomes the ghostproduct rule

χK χJ =

(−K − J−K

)χK+J , J ≥ 0. (5.6)

212 P.J. OLVER

The product makes sense as long as one of the multi-indices is non-negative,provided we adopt the Pochhammer definition(

J +K

J

)=

1

J !

p∏ν=1

jν−1∏i=0

(jν + kν − i) (5.7)

for the multinomial symbol. And, indeed, only such products will appearwhen we evaluate commutators and apply vector fields to nonlocal differ-ential polynomials.

The precise ghost calculus rules for computing the commutators ofghost characteristics will now be described. The commutators of ordinary

characteristics [Q,R] for Q,R ∈ B(∞)� follow the same rules (2.5) as in

the local case, where we replace the multiplication of monomials by theghost multiplication rule (5.6). Secondly, since ghosts do not involve thedependent variables, they mutually commute:

[χJ , χK ] = 0. (5.8)

Finally, the ghost characteristics χJ act as derivations on the ordinarycharacteristics:

[χK , QR] = Q χK(R) +R χK(Q).

Thus, we only need to know how to commute ghosts and derivative coor-dinates,

[χJ , uK ] = χJ+K (5.9)

in order to compute in the ghost characteristic space.

Example 5.5. Let us revisit Example 5.1. The three ghost characteristicsare

1 = χ0, ux = u1, D−1x u = u−1.

Then the three terms are

[χ0, [u1, u−1]] = 0, [u1, [u−1, χ0]] = −[u1, χ−1] = χ0,[u−1, [χ0, u1]] = [u−1, χ1] = −χ0.

The sum of these three terms is 0, and so the Jacobi paradox is resolved.

Example 5.6. The first Jacobi identity paradox that was found in [34],while working on the symmetry algebra of the KP equation, [9, 10, 15, 29],was more complicated than (3.1). Here p = 2, with independent variablesx, y, and q = 1, with dependent variable u. Consider the vector fields withcharacteristics y, y ux and uxD

−1x uy. As before, without the introduction

of ghost terms, the Jacobi sum

[y, [uxD−1x uy, yux]] + [yux, [y, uxD

−1x uy]] + [uxD

−1x uy, [yux, y]] (5.10)


equals − 2 y ux, not zero. In this case, the three ghost characteristics are

y = χ0,−1, yux = χ0,−1 u1,0, uxD−1x uy = u1,0 u−1,1.

Then,

[χ0,−1, χ0,−1 u1,0] = 2χ1,−2,[χ0,−1, u0,0u1,0] = χ0,−1 u1,0,

[χ0,−1 u1,0, u1,0 u−1,1] = Du1,0 u−1,1(χ0,−1 u1,0)− χ0,−1Dx(u1,0 u−1,1)= u0,0u1,0,

and so,

[u1,0 u−1,1, 2χ1,−2] = −2χ0,−1 u1,0,[u1,0 u−1,1, χ0,−1] = −χ−1,0 u1,0,

[χ0,−1 u1,0,−χ−1,0 u1,0] = χ0,−1 u1,0.

The latter three terms add up to 0, and so the Jacobi identity is valid inthe ghost framework.

6. Conclusions

In this paper, I have introduced a general framework for a nonlocal differen-tial algebra that will handle quite general nonlocal polynomial expressions.Several further topics of investigation are now of importance:

(a) A complete re-evaluation of earlier work on nonlocal symmetries oflocal and non-local partial differential equations is required. A properunderstanding of the hitherto undetected ghost terms needs to beproperly incorporated into earlier results, including the study of recur-sion operators and master symmetries, all of which typically involvenonlocal operations.

(b) The framework for the geometric and algebraic study of nonlocalsymmetries and nonlocal differential equations requires further devel-opment. The establishment of a complete nonlocal variational calculuson the nonlocal differential algebra Q(∞), including nonlocal conser-vation laws, [40] and a nonlocal form of Noether’s Theorem, [32, 33],would be a very worthwhile project for both theoretical developmentsand practical applications.

(c) Implementation of the nonlocal ghost calculus in standard computeralgebra packages would help a lot in these investigations.

214 P.J. OLVER

References

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2. Abraham–Shrauner, B., Hidden symmetries, first integrals and reduction of orderof nonlinear ordinary differential equations, J. Nonlinear Math. Phys. 9 Suppl. 2(2002), 1–9.

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INTEGRABLE BCN ANALYTIC DIFFERENCE OPERATORS:

HIDDEN PARAMETER SYMMETRIES

AND EIGENFUNCTIONS

S.N.M. RUIJSENAARSCentre for Mathematics and Computer Science,P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

Abstract. We consider integrable N-particle quantum systems of Calogero-Moser type,focusing on the ‘relativistic’ BCN setting, where commuting analytic difference operatorsarise. We show that the defining operators at the hyperbolic/elliptic levels, which dependon four/eight coupling constants, can be transformed to a manifestly D4/D8 symmetricform, resp. We survey various results on special eigenfunctions (including ‘ground states’)with regard to the latter symmetries and other ones. We also sketch a symmetry scenariofor the arbitrary-N eigenfunctions, motivated by the hyperbolic BC1 case, where our‘relativistic’ hypergeometric function has all of the expected properties.

1. Introduction

There exists a wide-spread belief that integrable N -particle quantum Ham-iltonians—especially those leading to factorized scattering—are highly ex-ceptional objects. We have recently shown that this is not the case [1], butin the following we focus on the quite special class of Calogero-Moser typeintegrable quantum systems, and more specifically on those connected tothe root system BCN .

We will detail the most general Hamiltonians of this family in themain text, whereas in this introduction we only present some illuminatingspecial cases. Moreover, we will be quite brief on background material,referring to the survey by Olshanetsky and Perelomov [2] and our lecturenotes [3] for extensive information on the ‘nonrelativistic’ Calogero-Mosersystems connected to root systems, and their ‘relativistic’ generalizations,resp. (To be precise, only for the root system AN−1 there is a naturalrelation to the nonrelativistic/relativistic space-time symmetry groups [3].For convenience, we use the same terminology for arbitrary root systems.)

217


218 S.N.M. RUIJSENAARS

As explained in [3], the relativistic setting gives rise to Hamiltoniansthat are commuting difference operators, whose step size is proportional to1/c, with c the speed of light. In the limit c→∞, these difference operatorsreduce to the commuting differential operators of the nonrelativistic case.

We recall that the defining AN−1 type differential operator is of theform

H(A)nr = −

N∑j=1

∂2j2+ g(g − 1)

∑1≤j<k≤N

V (xj − xk), (1.1)

with the most general V (x) being the Weierstrass ℘-function. For theremaining classical root systems BN , CN ,DN and BCN , a term of the form

g(g − 1)∑

1≤j<k≤NV (xj + xk) +

N∑j=1

U(xj) (1.2)

should be added to H(A)nr . For our purposes, the elliptic BCN case, where

one may choose

H(BC)nr = −

N∑j=1

∂2j2+ g(g − 1)

∑1≤j<k≤N

[V (xj − xk)

+ V (xj + xk)] +N∑j=1

3∑t=0

gt(gt − 1)2

V (xj + ωt), (1.3)

should be explicitly mentioned. This case was first studied by Inozemt-sev [4], and its integrability was shown by Oshima and H. Sekiguchi [5](cf. also [6]).

The above Hamiltonians have a manifest coupling constant symmetry:The pair potential part is invariant under g → 1− g and the external fieldpart under gt → 1−gt. A principal aim of this contribution is to reveal non-manifest symmetries of the relativistic/difference operator generalizationsof the above Hamiltonians, and to discuss their eventual relevance for thejoint eigenfunctions of the commuting difference operators.

The defining difference operator for the relativistic elliptic AN−1 casemay be chosen as [7]

H(A)rel =

N∑j=1

⎛⎝∏k �=j

σ(xj − xk + μ)σ(xj − xk)

⎞⎠12exp(iβ∂xj )⎛⎝∏l �=j

σ(xj − xl − μ)σ(xj − xl)

⎞⎠12+ (x→ −x), (1.4)

where σ(z) is the Weierstrass σ-function. Its BCN version was first pro-posed by van Diejen [8], who found some commuting Hamiltonians and

INTEGRABLE BCN ANALYTIC DIFFERENCE OPERATORS 219

conjectured the general form of the desired N commuting difference oper-ators. Subsequently, Hikami and Komori [9], in a tour de force of R-matrixtechnology, arrived at N commuting operators of the expected form.

The latter cannot easily be written down explicitly (by contrast tothe AN−1 case [7]). Even in the trigonometric/hyperbolic setting, the Ncommuting difference operators first found by van Diejen [10] have a quitecomplicated combinatorial structure. Our concern in this paper is primarilythe defining (‘simplest’) BCN Hamiltonian, both at the hyperbolic and atthe elliptic level.

In the trigonometric regime this Hamiltonian was first introduced byKoornwinder [11], who found and studied its polynomial eigenfunctions,following previous work by Macdonald. Specializing to the rank-one (N =1) case, these polynomials are the well-known Askey-Wilson polynomi-als [12, 13], which depend on four coupling type parameters. For N > 1the four parameters are present as well, together with an extra one. Thisfifth parameter plays the role of the coupling g in (1.3) and μ in (1.4),whereas the four parameters may be viewed as external field couplings,just as g0, . . . , g3 in (1.3).

In the hyperbolic regime the same number of parameters occurs, whereasin the elliptic regime one can allow eight parameters of external field cou-pling type. The hidden parameter symmetries alluded to above concern,first and foremost, certain non-manifest invariances w.r.t. the four/eightexternal field parameters in the hyperbolic/elliptic case. (Other non-obvioussymmetries are present as well, as discussed later on.) Specifically, we showthat the defining hyperbolic/elliptic difference operator has a D4/D8 sym-metry in the parameters. That is, it is not only invariant under arbitrarypermutations in S4/S8, but also under any even number of sign flips of theparameters.

As will be seen, the S4-invariance property in the hyperbolic case be-comes manifest when the four ‘physical’ couplings are traded for four shiftedparameters. (For the rank-one trigonometric case, the resulting Askey-Wilson operator is just the usual one [12, 13]; in this case the S4-invarianceof the suitably normalized polynomial eigenfunctions was already pointedout by Askey and Wilson [12].)

The S8-invariance at the elliptic level, however, is far from clear fromRefs. [8, 9], even for the N = 1 case. Indeed, van Diejen [8] works withshifted σ-functions σ(z + ωt), and Komori/Hikami [9] with theta functionsθs(z) (where t, s − 1 = 0, . . . , 3), using eight associated couplings whosepermutations give rise to manifestly distinct operators. Even so, also atthe elliptic level suitable shifts, alongside with multiplication by a (non-permutation-invariant!) constant, give rise to S8-invariance. To be sure,this follows only after substantial calculations.


We will therefore proceed differently. We start from difference operatorswhose S8-invariance is ‘built in’, and then show that they are in fact D8-invariant. In Appendix B we spell out how these operators relate to thoseof Refs. [8, 9]. The main tool for arriving at the manifestly symmetric ver-sions of the hyperbolic/elliptic operators is the hyperbolic/elliptic gammafunction from our paper [14], and therefore we begin by collecting somesalient features of these functions and related ones in Section 2.

In Section 3 we study the N = 1 Hamiltonians, with Subsections 3.1and 3.2 treating the hyperbolic and elliptic case, resp. Once the D4/D8

symmetry is well-understood for N = 1, the general-N case handled inSection 4 presents no new difficulties of principle. Here we also discussfurther properties that are relevant to the search for eigenfunctions.

Both in Section 3 and in Section 4 we need a few crucial functionalidentities that we collect in some lemmas. The proofs of these lemmas areof a calculational rather than a conceptual nature, and so we have relegatedthem to Appendix A. We would like to stress that in the elliptic case theseidentities amount to an adaptation of an N = 1 result due to van Diejen [8]and an arbitrary-N result due to Komori/Hikami [9], cf. also Appendix B.

In Section 5 we discuss eigenfunctions of the pertinent AΔOs with aneye on the parameter symmetries at issue, and also with regard to theirsuitability for Hilbert space purposes. Subsection 5.1 may be viewed as anintroduction and overview of various aspects. It also contains remarks ona possible scenario for the arbitrary N , arbitrary coupling and arbitraryeigenvalue eigenfunctions that are yet to be constructed/discovered.

In Subsection 5.2 we have tried to sketch in a discursive fashion variousresults from our papers [15–17], which deal with a ‘relativistic’ hypergeo-metric function. This is an eigenfunction of the hyperbolic BC1 AΔO, andwe focus on features that may be illuminating as a guide for the generalcase. In particular, the character and occurrence of a ‘ground state’ canbe completely controlled in the hyperbolic BC1 setting. This ground stateissue is studied for the general case in Subsection 5.3.

Section 6 contains information regarding the nonrelativistic limit. Inparticular, we detail how the defining AΔOs reduce to differential operators,and we clarify the fate of the various symmetries under this limit. In theprocess, we obtain the limits of the eigenfunctions of ground state type. Atthe end of Section 6 we comment on the connection of the latter functionswith recent work on quasi-exactly solvable models by the Madrid school [18,19].

Appendix B is concerned with the precise relation of our form of theelliptic BCN operators and the one found in the papers by van Diejen [8]and Komori/Hikami [9].


2. Preliminaries

In this section we collect known results on the hyperbolic and ellipticgamma functions, including some formulas relating to elliptic and alliedfunctions. For more information and background material we refer to [20,14, 21].

To begin with, throughout this paper we use parameters a+, a−, r thatare positive. It is convenient to set

a ≡ (a+ + a−)/2, α ≡ 2π/a+a−. (2.1)

With these conventions, the hyperbolic gamma function can be defined bythe integral representation

G(a+, a−; z) = exp

(i

∫ ∞

0

dy

y

(sin 2yz

2 sinh(a+y) sinh(a−y)− z

a+a−y

)),

|Im z| < a. (2.2)

It extends to a meromorphic function satisfying the analytic differenceequation (AΔE)

G(z + ia+/2)

G(z − ia+/2)= 2 cosh(πz/a−). (2.3)

Up to normalization, it is the unique minimal solution to this AΔE, in asense explained in [14].

The symmetry under a+ ↔ a− that is manifest from (2.2) entails thatG(a+, a−; z) also obeys the AΔE

G(z + ia−/2)G(z − ia−/2)

= 2 cosh(πz/a+). (2.4)

The only further properties of G we invoke in this paper are the reflectionequation

G(a+, a−;−z) = 1/G(a+, a−; z), (2.5)

(which is clear from (2.2)), the duplication formula

G(a+, a−; 2z) =∏

l,m=+,−G(a+, a−; z + i(la+ +ma−)/4), (2.6)

cf. [14] (3.24)–(3.25), and the limits

lima−↓0

G(a+, a−; z − ia−κ)G(a+, a−; z − ia−λ)

= exp[(λ− κ) ln(2 cosh(πz/a+))], (2.7)

where κ, λ ∈ IR, and z belongs to the cut planeC| \{±i[a+/2,∞)} (cf. [14] (3.91)),

lima−↓0

G(a+, a−; ia+/2− ia−κ) exp[κ ln(2πa−/a+)] =(2π)1/2

Γ(κ+ 1/2), (2.8)


where κ ∈ C| , cf. Proposition III.6 in [14].The elliptic gamma function depends not only on the parameters a+, a− >

0, but also on a parameter r > 0. It can be defined by the infinite product

G(r, a+, a−; z) =∞∏

m,n=1

1− q2m−1+ q2n−1− e−2irz

1− q2m−1+ q2n−1− e2irz, qδ ≡ exp(−raδ). (2.9)

From this it is clear that G is meromorphic in z, with zeros and poles thatcan be read off.

Defining the ‘right-hand-side function’

R(r, a; z) =∞∏k=1

(1− exp[2irz − (2k − 1)ra])(z → −z), a > 0, (2.10)

it is readily verified that G solves the AΔEs

G(z + iaδ/2)

G(z − iaδ/2)= R(r, a−δ; z), δ = +,−. (2.11)

Up to normalization, our elliptic gamma function is the unique minimalsolution to each of these AΔEs. (Here we think of R as a right-hand-sidefunction given by (2.10).)

We will have occasion to invoke the reflection equation

G(r, a+, a−;−z) = 1/G(r, a+, a−; z), (2.12)

and the duplication formula

G(r, a+, a−; 2z) =∏

l,m=+,−G

(r, a+, a−; z −

i

4(la+ +ma−)

)

×G(r, a+, a−; z −

i

4(la+ +ma−)−

π

2r

), (2.13)

cf. [14] (3.106). Another feature that is needed is the alternative represen-tation

G(r, a+, a−; z) = exp

(i∞∑n=1

sin 2nrz

2n sinh(nra+) sinh(nra−)

), |Im z| < a.

(2.14)It entails in particular the hyperbolic limit

limr↓0G(r, a+, a−; z) exp(π2z/6ira+a−) = G(a+, a−; z), (2.15)

cf. Proposition III.12 in [14]. We also need another r ↓ 0 limit that cannotbe found in [14], namely

limr↓0G(r, a+, a−; z + π/2r) exp(−π2z/12ira+a−) = 1, z ∈ C| . (2.16)


But this limit easily follows from (2.14) and (2.15). (See also the proof ofProposition III.12 in [14].) Finally, we have occasion to invoke the limit

lima−↓0

G(r, a+, a−; z − ia−κ)G(r, a+, a−; z − ia−λ)

= exp[(λ− κ) ln(R(r, a+; z))], (2.17)

with κ, λ ∈ IR, and z in the cut plane C| \ ∪k∈ZZ{kπ/r ± i[a+/2,∞)},cf. [14] (3.138).

We now turn to properties of R(r, a; z) and related functions. To startwith, it is plain from the product representation (2.10) that R(r, a; z) is aneven, π/r-periodic, entire function of z, which satisfies

R(r, a; z + ia/2)

R(r, a; z − ia/2) = − exp(−2irz), (2.18)

and which has simple zeros for z = kπ/r + i(l + 1/2)a, with k, l ∈ ZZ.Combining (2.13) and (one of) the AΔEs (2.11), we obtain the duplicationformula

R(r, a; 2z) = R(r, a; z + ia/4)R(r, a; z − ia/4)R(r, a; z + ia/4 − π/2r)×R(r, a; z − ia/4− π/2r). (2.19)

Likewise, (2.14) entails the representation

R(r, a; z) = exp

(−

∞∑n=1

cos 2nrz

n sinh(nra)

), |Im z| < a/2, (2.20)

and (2.15)–(2.16) and (2.11) yield the limits

limr↓0

exp(π2/6ra)R(r, a; z) = 2 cosh(πz/a), (2.21)

limr↓0

exp(−π2/12ra)R(r, a; z + π/2r) = 1. (2.22)

Another convenient function can be defined by

s(r, a; z) ≡ ie−irzR(r, a; z − ia/2)/2r∞∏k=1

(1− pk)2, p ≡ e−2ra. (2.23)

The product representation

s(r, a; z) =sin rz

r

∞∏k=1

(1− pke2irz)(z → −z)(1− pk)2 (2.24)

following from (2.10) shows that we have

s(r, a; z) = exp(−ηz2r/π)σ(z;π/2r, ia/2), (2.25)


where σ is the Weierstrass σ-function.We continue by tying in our functions s(r, a; z) and R(r, a; z) with the

shifted σ-functions and θ-functions employed by van Diejen [8] and Ko-mori/Hikami [9], resp. (The remainder of this section is invoked only inAppendix B.) To this end we set

ω0 = 0, ω1 = π/2r, ω2 = ia/2, ω3 = −ω1 − ω2, (2.26)

η0 = 0, ηt = ζ(ωt), t = 1, 2, 3, (2.27)

where ζ is the Weierstrass ζ-function. Now van Diejen works with the fourfunctions

σ0(z) ≡ σ(z), σt(z) ≡ exp(−ηtz)σ(z + ωt)/σ(ωt), t = 1, 2, 3. (2.28)

Their connection to s and R can be made via (2.25) and (2.23).The θ-functions θs(z) in Eqs. (A.1a)–(A.1d) of [9] are related to the

θ-functions θWWs (z) from [20] by

θs(z) = θWWs (πz), s = 1, . . . , 4. (2.29)

Their period lattice is ZZ+τZZ. Hence our parameters r, a should be chosenas

r = π, a = −iτ. (2.30)

Putting

q ≡ exp(iπτ) = exp(−πa), G ≡∞∏k=1

(1− q2k), (2.31)

the product formula (2.24) with r = π and p = q2 can be compared to theproduct formula for θ1(z) (cf. p. 470 in [20]), to obtain the relation

θ1(z) = 2πq1/4G3s(π, a; z). (2.32)

In particular, this entails

θ′1(0) = 2πq1/4G3. (2.33)

Next, comparing (2.10) to p. 469 in [20], we deduce

θ4(z) = GR(π, a; z). (2.34)

The relation of our functions s and R to the remaining θ-functions

θ2(z) ≡ θ1(z + 1/2), θ3(z) ≡ θ4(z + 1/2), (2.35)

is then clear. Finally, for later use we note that (2.23) entails

s(π, a; z) = (−2iπG2)−1 exp(−iπz)R(π, a; z − ia/2). (2.36)


3. The BC1 Hamiltonians

The hyperbolic/elliptic operators depend on four/eight parameters hn, withn = 0, . . . , 3/7, resp. Unless specified otherwise, the hn’s are arbitrarycomplex numbers. A key role in both regimes is played by the c-function

c(z) ≡ 1

G(2z + ia)

∏n

G(z − hn), (3.1)

where G is the hyperbolic/elliptic gamma function and the product is overn = 0, . . . , 3/n = 0, . . . , 7 in the hyperbolic/elliptic case. (Here and in thesequel, we often suppress the dependence on r, a+, a− and various otherparameters, whenever the context is such that no confusion is likely toarise.) Two other functions occurring in both cases are the w-function

w(z) ≡ 1/c(z)c(−z), (3.2)

and the u-function

u(z) ≡ − exp(−2irz)c(z)/c(−z). (3.3)

(Of course, r = 0 in the hyperbolic case.) Note that all of these functions areinvariant under interchange of a+ and a− and under arbitrary permutationsof the parameters hn. Using the reflection equations (2.5) and (2.12), theu-function can be rewritten as

u(z) = − exp(−2irz)∏nG(z − hn)G(z + hn)G(2z + ia)G(2z − ia) . (3.4)

Hence it is also invariant under arbitrary sign flips of hn.Consider now the coefficient function

Va(z) ≡ exp(−2ra−)u(z + ia−)/u(z). (3.5)

Obviously, the z-shift breaks the a+ ↔ a− symmetry. But Va is still invari-ant under arbitrary permutations and sign flips of hn. The defining BC1

analytic difference operator (AΔO) can now be chosen to be of the form

A = exp(−ia−∂z) + Va(z) exp(ia−∂z) + Vb(z). (3.6)

As will be detailed shortly, the additive ‘potential’ Vb(z) is even in z andpermutation invariant in the parameters hn, but no longer invariant underarbitrary sign flips. It is however invariant under sign flips that involvean even number of parameters. Thus we arrive at the above-mentionedD4/D8-invariance.


There are two similarity transforms of A that are also crucial. First, wecan transform A with the c-function, yielding

A ≡ c(z)Ac(z)−1= V (z) exp(−ia−∂z) + V (−z) exp(ia−∂z) + Vb(z), (3.7)

withV (z) ≡ c(z)/c(z − ia−). (3.8)

Note this entails the relation

Va(z) = V (−z)V (z + ia−). (3.9)

The second transformation is given by

H ≡ (−e2irzu(z))1/2A(−e2irzu(z))−1/2. (3.10)

Thus H can be rewritten as

H = V (z)1/2 exp(−ia−∂z)V (−z)1/2 + (z → −z) + Vb(z), (3.11)

or asH = Va(z)

1/2 exp(ia−∂z) + (z → −z) + Vb(z). (3.12)

Moreover, we haveH = w(z)1/2Aw(z)−1/2, (3.13)

as is easily verified.On first acquaintance, having three choices for a defining operator may

seem strange. Therefore we add some brief comments. To begin with, theHamiltonian H in the form (3.11) arises most naturally from quantizationof a classical version, cf. (1.4) and [3]. In the representation (3.11) (to-gether with the definition of Vb(z) detailed below), permutation invarianceis plain, but D4/D8 invariance is non-obvious. Even so, H is in fact D4/D8-symmetric, as follows from the D4/D8-invariance of Va(z) and Vb(z). Theoperator A, however, is only permutation invariant.

Imposing suitable restrictions on the parameters hn (specified below),the u-function is a phase for real z, and H is a formally hermitian opera-tor on L2(IR, dz) that commutes with parity. Thus the operator A is alsoformally hermitian on L2(IR, dz), but it no longer commutes with parity.Furthermore, w(z) is positive for real z and the operator A is formallyhermitian on L2(IR,w(z)dz); since w(z) is even and A commutes withparity, it leaves even and odd functions invariant.

To be sure, this appraisal of the most conspicuous differences betweenthe three versions is quite sketchy even at a formal level. Later on, we willsay more about the pros and cons of the three choices, in connection with


known and ‘expected’ Hilbert space eigenfunction transforms. But now weproceed by zooming in on the hyperbolic and elliptic cases.

3.1. THE HYPERBOLIC REGIME: D4 SYMMETRY

From now on we work with the functions

sδ(z) ≡ sinh(πz/aδ), cδ(z) ≡ cosh(πz/aδ), δ = +,−. (3.14)

Using the definitions (3.8) and (3.1), and then the AΔE (2.4), we readilycalculate

V (z) = −4∏n c+(z − hn − ia−/2)s+(2z)s+(2z − ia−)

. (3.15)

Hence (3.9) yields

Va(z) =16

∏n c+(z + hn + ia−/2)c+(z − hn + ia−/2)s+(2z)s+(2z + ia−)2s+(2z + 2ia−)

. (3.16)

Clearly, (3.16) exhibits manifest invariance under arbitrary sign flips ofhn. Likewise, (3.15) shows V (z) is not invariant under any flips (for generich = (h0, . . . , h3), of course).

In the hyperbolic case the function Vb(z) is defined by

Vb(z) ≡ −V (z)− V (−z)− 2c+(∑hn + ia−). (3.17)

Thus Vb(z) is S4-invariant, just as V and Va. The additive constant ensuresthe asymptotics

Vb(z)→ 0, |Re z| → ∞, (3.18)

as is easily verified.The point is now that Vb(z) admits a second representation that reveals

it is also invariant under an even number of sign flips; hence it is (bydefinition) D4-invariant.

Lemma 3.1 We have

Vb(z) =2pc

c+(z − ia)c+(z + ia)− 2pss+(z − ia)s+(z + ia)

, (3.19)

where

pc ≡3∏n=0

c+(hn), ps ≡3∏n=0

s+(hn). (3.20)


We prove this lemma in Appendix A. Note that pc is invariant underarbitrary sign flips; in contrast, ps is only invariant under an even numberof flips.

Next, we introduce couplings c = (c0, . . . , c3) by setting

h0 = i(c0 − a), h1 = i(c1 − a−/2), h2 = i(c2 − a+/2), h3 = ic3. (3.21)Then we can rewrite (3.15) as

V (z) =s+(z − ic0)s+(z)

c+(z − ic1)c+(z)

s+(z − ic2 − ia−/2)s+(z − ia−/2)

× c+(z − ic3 − ia−/2)c+(z − ia−/2)

. (3.22)

From this the terminology ‘couplings’ can be understood: for cn = 0 thecorresponding ‘interaction term’ becomes 1. In particular, for c = 0, wehave

V (z) = 1, Vb(z) = 0, Va(z) = 1, (c = 0), (3.23)

so that

A = A = H = exp(−ia−∂z) + exp(ia−∂z), (c = 0). (3.24)

In words, the three versions of the BC1 AΔO reduce to the same ‘free’AΔO. Note also that the c-function (3.1) reduces to 1 for c = 0, due to theduplication formula (2.6).

On the one hand, the parametrization of the AΔO-coefficients via thecouplings c is natural from a physical viewpoint, as just illustrated. On theother hand, it breaks the above symmetries, in the sense that e.g. V (z)(3.22) is no longer invariant under any permutation of c0, . . . , c3. In termsof c, therefore, even permutation invariance is ‘hidden’. We will return tothis issue in Subsection 5.2, where we review our BC1-eigenfunction.

Finally, we observe that when we choose c ∈ IR4 and z ∈ IR, then u(z)is a phase, w(z) is positive, Vb(z) is real-valued, and the complex conjugateof V (z) equals V (−z), cf. (3.22). Hence H and A are formally self-adjointon L2(IR, dz), and A is formally self-adjoint on L2(IR,w(z)dz).

3.2. THE ELLIPTIC REGIME: D8 SYMMETRY

Proceeding along the same lines as in the hyperbolic case, we begin byintroducing the notation

sδ(z) ≡ s(r, aδ ; z), Rδ(z) ≡ R(r, aδ ; z), δ = +,−. (3.25)

Now we use (3.8), (3.1) and (2.4) to express V (z) in terms of R+. Thisyields

V (z) =

∏nR+(z − hn − ia−/2)

R+(2z + ia+/2)R+(2z − ia− + ia+/2). (3.26)


From (3.9) we then get

Va(z) = (3.27)∏nR+

(z + hn +

i2a−

)R+(z − hn + i

2a−)

R+

(2z − i

2a+)∏

δ=+,−R+

(2z + ia− + i

2δa+)· R+

(2z + 2ia− + i

2a+) .

Just as in the hyperbolic case, we see that Va(z) is invariant underarbitrary sign flips of hn, whereas V (z) is not invariant. The obvious analogof the rhs of (3.17) reads−V (z)−V (−z)+C, where C is a suitable constant.But as will become clear shortly, this is not a D8-invariant function. On theother hand, with a certain restriction on h = (h0, . . . , h7) in effect, Vb(z) isindeed of this form. For general h, however, the definition of Vb(z) is quitedifferent, and yields a function that is D8-invariant.

Admittedly, at this stage the definition of Vb(z) on which we now embarkmay appear ad hoc. Its motivation arises from the N > 1 generalization,cf. Section 4 and Appendix B. First, we introduce half-periods ωt, t =0, . . . , 3:

ω0 = 0, ω1 = π/2r, ω2 = ia+/2, ω3 = −ω1 − ω2. (3.28)

Second, we introduce four functions that involve a new parameter μ ∈ C| :

Et(μ; z) ≡R+(z + μ− ia− ωt)R+(z − μ+ ia− ωt)

R+(z − ia− ωt)R+(z + ia− ωt), (3.29)

t = 0, . . . , 3.

Third, we define the product functions

p0 ≡∏n

R+(hn), p1 ≡∏n

R+(hn − ω1), (3.30)

p2 ≡ exp(−2ra+)∏n

exp(−irhn)R+(hn − ω2), (3.31)

p3 ≡ exp(−2ra+)∏n

exp(irhn)R+(hn − ω3). (3.32)

Finally, we introduce

Vb(z) ≡∑3

t=0pt[Et(μ; z)− Et(μ; zt)]

2R+(μ− ia+/2)R+(μ− ia− − ia+/2), (3.33)

with

z0 = z2 = ω1, z1 = z3 = 0. (3.34)


In the next lemma we show in particular that −Vb(z) only differs fromthe function

Σ(z) ≡ V (z) + V (−z) (3.35)

by a constant, provided we have

i∑n

hn = 4a, (mod iπ/r). (3.36)

To appreciate this restriction, we first note that by virtue of (2.18) thefunctions Et(z) and Vb(z) are elliptic with periods π/r, ia+, whereas

V (z + ia+)/V (z) = exp(2r[i∑hn − 4a]). (3.37)

From (3.9) we then get

Va(z + ia+)/Va(z) = 1, (3.38)

so that Va(z) is elliptic, too. But V (z) is not ia+-periodic, unless (3.36)holds true. We are now prepared for the lemma, which is proved in Ap-pendix A.

Lemma 3.2 The function Vb(z) does not depend on μ. Moreover, with(3.36) in effect, the function

Σ(z) + Vb(z) (3.39)

does not depend on z.

The presence of the extra parameter μ anticipates the arbitrary-N gen-eralization, cf. the next section. In this connection we note the μ = 0representation

Vb(z) =ρ

2R+(ia− + ia+/2)(3.40)

×3∑t=0

pt([L+(z − ia− ωt)− L+(z + ia− ωt)]− [z → zt]),

whereρ ≡ Res(1/R+(z))|z=−ia+/2, (3.41)

and L+(z) is the logarithmic derivative

L+(z) ≡ R′+(z)/R+(z). (3.42)

As in the hyperbolic case, D8-invariance of the BC1 AΔO A (3.6)is now easily established. Indeed, the products p0, . . . , p3 are manifestly


permutation invariant. Since R+(z) is even and π/r-periodic, p0 and p1 areinvariant under arbitrary sign flips. Using also (2.18), we see that p3 hasthis feature, too, whereas p2 is only invariant under an even number of flips.

The upshot is that p0, . . . , p3 are D8-invariant. Since Va(z) is also in-variant, we have now shown that A is D8-invariant, as announced.

We continue by defining new parameters c0, . . . , c7 of coupling constanttype. Specifically, c0, . . . , c3 are defined as in the hyperbolic case, cf. (3.21),while c4, . . . , c7 are given by

h4 = i(c4 − a)− ω1, h5 = i(c5 − a−/2) − ω1,h6 = i(c6 − a+/2) + ω1, h7 = ic7 + ω1.

(3.43)

Using the duplication formula (2.19) we now obtain

V (z) =R+(z − ic0 + ia+/2)R+(z + ia+/2)

R+(z − ic1)R+(z)

(3.44)

× R+(z − ic2 + i(a+ − a−)/2)R+(z + i(a+ − a−)/2)

R+(z − ic3 − ia−/2)R+(z − ia−/2)

× R+(z − ic4 + ia+/2 + ω1)R+(z + ia+/2 + ω1)

R+(z − ic5 + ω1)R+(z + ω1)

× R+(z − ic6 + i(a+ − a−)/2 + ω1)R+(z + i(a+ − a−)/2 + ω1)

R+(z − ic7 − ia−/2 + ω1)R+(z − ia−/2 + ω1)

,

showing the coupling character of cn. Trading h for c in the productsp0, . . . , p3 leads to unilluminating formulas, however. Notice that the re-striction (3.36) is equivalent to∑

n

cn = 0, (mod π/r). (3.45)

Obviously, for c = 0 all of the products pt vanish, so that Vb(z) vanishesfor zero coupling. More generally, we obtain once again (3.23) and (3.24),and the c-function reduces to 1 for c = 0 (by the duplication formula(2.13)).

It is illuminating to study the hyperbolic limit r ↓ 0, with the couplingsc0, . . . , c7 fixed. Using (2.21) and (2.22), we get from (3.44)

limr↓0V (z) =

c+(z − ic0 + ia+/2)c+(z + ia+/2)

c+(z − ic1)c+(z)

(3.46)

× c+(z − ic2 + i(a+ − a−)/2)c+(z + i(a+ − a−)/2)

c+(z − ic3 − ia−/2)c+(z − ia−/2)

.

This equals the hyperbolic coefficient V (z) (3.22). From (2.21) and (2.22)we also deduce that Vb(z) has a finite limit. Specifically, we obtain

limr↓0Vb(z) =

2

s+(μ)s+(ia− − μ)(pc[Qc(z) − 1] + ps[Qs(z)− 1]), (3.47)


where

Qc(z) ≡c+(z + μ− ia)c+(z − μ+ ia)

c+(z − ia)c+(z + ia),

Qs(z) ≡s+(z + μ− ia)s+(z − μ+ ia)

s+(z − ia)s+(z + ia), (3.48)

and pc, ps are given by (3.20).Since Vb(z) does not depend on μ, the same is true for the function on

the rhs of (3.47). Indeed, a straightforward calculation shows that it equalsthe hyperbolic Vb(z), as given by (3.19).

We conclude this subsection by observing that the choices c ∈ IR8, z ∈IR, entail that u(z) is a phase, w(z) is positive, Vb(z) is real-valued, and V (z)equals V (−z), cf. (3.44). It readily follows that H and A are formally self-adjoint on L2((−π/2r, π/2r), dz) and A on L2((−π/2r, π/2r), w(z)dz). (Ofcourse, working formally, we may as well choose IR instead of (−π/2r, π/2r),but the latter choice is more natural from a physical viewpoint.)

4. The BCN case

In this section we proceed along the same lines as in the previous one. Thuswe begin by specifying the features of the BCN AΔOs that are shared bythe hyperbolic and elliptic versions. To complete the definition of the AΔOs,however, we need to consider the hyperbolic and elliptic regimes separately.

Apart from the ‘external field’ c-function (3.1), we need a second c-function

cμ(z) ≡G(z − μ+ ia)G(z + ia)

, μ ∈ C| , (4.1)

as building block. (As before, G denotes the elliptic or hyperbolic gammafunction.) We now define factorized C-functions depending on N complexvariables x = (x1, . . . , xN ), namely the external field C-function

Ce(x) ≡N∏j=1

c(xj), (4.2)

the sum and difference C-functions

C±(x) ≡∏

1≤j<k≤Ncμ(xj ± xk), (4.3)

and the total C-function

C(x) ≡ Ce(x)C+(x)C−(x). (4.4)


Finally, we define a weight function

W (x) ≡ 1/C(x)C(−x). (4.5)

The three versions of the defining BCN AΔO are now given by

H ≡N∑j=1

(Vj(x)1/2 exp(−ia−∂xj )Vj(−x)1/2 + (x→ −x)) + V(x), (4.6)

A ≡W (x)−1/2HW (x)1/2, (4.7)

A ≡ C(x)−1AC(x). (4.8)

The coefficients Vj(x) can be written

Vj(x) ≡ V(s)j (x)V(A)j (x). (4.9)

Here, the ‘simple’ factor is of the form

V(s)j (x) ≡ V (xj)C+(x)/C+(x− ia−ej), (4.10)

where V (z) is given by (3.8), and e1, . . . , eN is the canonical basis of C| N .

The remaining (AN−1-type) factor V(A)j can also be specified by one formulaapplying at once to the hyperbolic and elliptic cases, but only at the expenseof introducing a function

f(z) ≡{2c+(z), (hyperbolic case),R+(z), (elliptic case).

(4.11)

Then it reads

V(A)j (x) ≡∏k �=j

f(xj − xk − μ+ ω2)f(xj − xk + ω2)

, ω2 = ia+/2. (4.12)

We note that in terms of f we also have

V(s)j (x) =

∏n f(xj − hn − ia−/2)

f(2xj + ω2)f(2xj − ia− + ω2)·∏k �=j

f(xj + xk − μ+ ω2)f(xj + xk + ω2)

. (4.13)

(Recall (3.15) and (3.26) to see this.)Before turning to the definition of V(x), we use the above factorizations

to calculate the shift coefficients in A and A. First, we assert that we have

A =N∑j=1

(Vj(x) exp(−ia−∂j) + Vj(−x) exp(ia−∂j)) + V(x). (4.14)


Indeed, this amounts to the identities

Vj(x) =C(x)C(−x)

C(x− ia−ej)C(−x+ ia−ej)Vj(−x+ ia−ej), j = 1, . . . , N.

(4.15)To prove these, we note that (4.10) and (3.8) entail that (4.15) with Vj →V(s)j and C → CeC+ is clearly true. Thus we need only show

V(A)j (x) =C−(x)C−(−x)

C−(x− ia−ej)C−(−x+ ia−ej)V(A)j (−x+ ia−ej). (4.16)

To this end we calculate the functions

V−j (x) ≡ V(A)j (x)C−(x− ia−ej)/C−(x), j = 1, . . . , N. (4.17)

Using (4.3), (4.1) and the G-AΔEs, we obtain

C−(x− ia−ej)C−(x)

=∏k>j

cμ(xj − xk − ia−)cμ(xj − xk)

·∏l<j

cμ(xl − xj + ia−)cμ(xl − xj)

=∏k>j

f(xj − xk + ω2)f(xj − xk − μ+ ω2)

·∏l<j

f(xl − xj − μ+ ia− + ω2)f(xl − xj + ia− + ω2)

. (4.18)

Multiplying by (4.12), we now get

V−j (x) =∏l<j

f(xj − xl − μ+ ω2)f(xj − xl + μ− ia− − ω2)f(xj − xl + ω2)f(xj − xl − ia− − ω2)

. (4.19)

Finally, we observe that the rhs is invariant under x → −x+ ia−ej (sincef is even), and so (4.16) follows.

We have now proved the representation (4.14) for A. Our calculationsalso entail a more explicit formula for A (4.8), viz.,

A =N∑j=1

(V−j (x) exp(−ia−∂j) + V+j (x) exp(ia−∂j)) + V(x), (4.20)

with V−j given by (4.19) and V+j by

V+j (x) = Va(xj)∏k �=j

f(xj + xk + μ− ω2)f(xj + xk − μ+ ia− + ω2)f(xj + xk − ω2)f(xj + xk + ia− + ω2)

·∏l>j

f(xj − xl + μ− ω2)f(xj − xl − μ+ ia− + ω2)f(xj − xl − ω2)f(xj − xl + ia− + ω2)

. (4.21)

Before presenting the definition of V(x), we connect the above devel-opments to the AN−1 Hamiltonian H

(A)rel (1.4) with β → a− and σ-periods


π/r, ia+. Consider H (4.6) with V(x) = 0 and Vj(x) replaced by V(A)j (x).Then it readily follows from (4.12) with f = R+ that H reduces to amultiple of (1.4). (Recall (2.18), (2.23) and (2.25) to see this.) Transforming

with C−(x) and W−(x) ≡ 1/C−(x)C−(−x) then yields AN−1 AΔOs A(A)rel

and A(A)rel with meromorphic coefficients that are clear from the above.

4.1. THE HYPERBOLIC AΔOS

We now specialize to the hyperbolic case, completing our AΔO definitionsby setting

V(x) ≡ 2

s+(μ)s+(ia− − μ)

⎛⎝pc[ N∏j=1

Qc(xj)− 1] + ps[N∏j=1

Qs(xj)− 1]⎞⎠ ,(4.22)

cf. (3.20), (3.48). As we have seen below (3.48), this equals Vb(x1) for N = 1.But for N > 1 we do get dependence on μ. (Take e.g. Reμ → ∞ to seethis.) On the other hand, we have

limμ→0V(x) =

N∑j=1

Vb(xj), (4.23)

as is easily verified, cf. (3.19). We also obtain the limit

V(x)→ 0, |Re x1|, . . . , |Re xN | → ∞. (4.24)

At first sight, V(x) seems to have no relation to the shift coefficientsVj(±x). However, introducing

Σ(x) ≡N∑j=1

(Vj(x) + Vj(−x)), (4.25)

we have the following lemma, which we prove in Appendix A.

Lemma 4.1 The function Σ(x) + V(x) does not depend on x. Explicitly,we have

V(x) = −Σ(x)− 2s+(Nμ)s+(μ)

c+

(3∑n=0

hn + (N − 1)μ+ ia−). (4.26)

From the above it is again clear that the AΔO A is D4-invariant. SinceC(x)/C(−x) is also D4-invariant, the Hamiltonian H is D4-invariant aswell. It should be noted that when V(x) is replaced by −Σ(x), then we


obtain AΔOs that are no longer D4-invariant. (Indeed, the constant on therhs of (4.26) is only S4-invariant.)

Another involutive symmetry of A is easily checked: it is invariant under

μ $→ 2ia− μ. (4.27)

Once more, this invariance is present for H, too, but not for A. (Indeed,the C-functions C±(x) are not invariant under (4.27).) On the other hand,A is only invariant under permutations of x1, . . . , xN , whereas A and H arealso invariant under arbitrary sign changes. (Thus the latter operators areinvariant under the BCN Weyl group.)

Just as for N = 1, we can introduce couplings c via (3.21). To obtaina positive weight function W (x) on IRN , we should not only take c ∈ IR4,but also choose τ ∈ IR, where τ is defined by

μ = iτ. (4.28)

With these reality conditions in effect, we readily obtain formal self-adjointnessof H and A on L2(IRN , dx), and of A on L2(IRN ,W (x)dx). In order to takeBCN -invariance into account, it is however more natural to replace IR

N bythe ‘Weyl chamber’

GN ≡ {x ∈ IRN | 0 < xN < · · · < x1}, (4.29)

cf. also the next section.

4.2. THE ELLIPTIC AΔOS

The elliptic additive potential V(x) is given by

V(x) ≡

∑3

t=0pt

⎛⎝ N∏j=1

Et(xj)− Et(zt)N⎞⎠

2R+(μ− ia+/2)R+(μ− ia− − ia+/2). (4.30)

Thus we obtain Vb(x1) for N = 1, cf. (3.33). Moreover, the limit (4.23) isreadily checked in this case, too (cf. the representation (3.40)–(3.42)).

As before, it now follows that A and H are D8-invariant, whereas A isonly S8-invariant. Moreover, A and H are invariant under the μ-involution(4.27), in contrast to A. Finally, A and H are BCN -invariant, whereas Ais invariant under arbitrary permutations of x1, . . . , xN , but not under signchanges.

We proceed with some comments that apply to the hyperbolic case aswell. The symmetry properties just discussed show that the operator H isthe ‘maximally symmetric’ one among the three versions. Its drawback is


the occurrence of square-root branch points in its coefficients. As a conse-quence, there is at first no natural choice for a function space on which itacts. By contrast, the AΔOs A and A have meromorphic coefficients, sothey have an obvious action on the space M of meromorphic functions.In view of the similarity transformations between H,A and A, the obviouschoice for a function space on which H acts is therefore the space

MC ≡ (C(x)/C(−x))1/2M =W (x)1/2M. (4.31)

The additive potential V(x) is not only meromorphic, but also ellipticin x1, . . . , xN with periods π/r, ia+. (Indeed, the functions E0(z), . . . , E3(z)are elliptic.) Likewise, the coefficients V±j (x) of the shifts in A are readily

seen to be elliptic, cf. (4.19)–(4.21). Consider now the AΔOs A, H and Athat result from A,H and A by interchanging a+ and a− in all of the abovedefinitions. (In particular, therefore, all coefficients of the new AΔOs areexpressed in terms of the functions c−, s− and R−.) Due to the ellipticityproperties of the A- and A-coefficients, it is immediate that we have

[A, A] = 0. (4.32)

That is, A and A commute as operators on M. Thus this is also true forthe operators A and A on M, and for H and H on MC (4.31). (Indeed,C(x) is invariant under a+ ↔ a−, since the hyperbolic and elliptic gammafunctions are.) Hence it is natural to try and find eigenfunctions in thesespaces that are also invariant under interchange of a+ and a−; we will returnto this issue in the next section.

In contrast to the coefficients of A, the functions Vj(±x) (4.9) appearingin A are generically not ia+-periodic. Indeed, for ia+-periodicity we shouldrequire

i∑n

hn + 2i(N − 1)μ = 4a, (mod iπ/r). (4.33)

(Use (4.11)–(4.13) and (2.18) to check this.) As in the hyperbolic case,we now define a sum function Σ(x) by (4.25). Then we have the followinglemma, whose proof is relegated to Appendix A.

Lemma 4.2 With the restriction (4.33) in force, the function Σ(x)+V(x)does not depend on x.

When we trade h for couplings c (cf. (3.21) and (3.43)), we see that for cand μ fixed the limit r ↓ 0 leads from the elliptic AΔOs to their hyperboliccounterparts. (This follows in the same way as in the N = 1 case from thetwo R+-limits (2.21), (2.22).)

Keeping r > 0 and choosing c ∈ IR8, τ ∈ IR (with τ defined by(4.28)), we obtain a positive weight function W (x) on IRN . In view of


BCN -invariance, the natural Hilbert spaces for H,A and A are L2(GN , dx)and L2(GN ,W (x)dx), resp., where GN is the BCN ‘Weyl alcove’

GN ≡ {x ∈ IRN | 0 < xN < · · · < x1 < π/2r}. (4.34)

The reality conditions just detailed give rise to formally self-adjoint opera-tors, as in the previous cases.

5. Eigenfunctions vs. parameter symmetries

5.1. INTRODUCTORY REMARKS

Compared to eigenfunctions of differential operators (for which abundantexistence and uniqueness results exist), very little is known about eigenfunc-tions of analytic difference operators. We have commented on the generaltheory at previous occasions [3, 22, 23]. In this section our emphasis is onspecial eigenfunctions of the above AΔOs, paying particular attention totheir parameter symmetries and quantum mechanical relevance.

More specifically, in Subsection 5.2 we summarize various results fromour papers [15–17] that have a bearing on these issues for the hyperbolicBC1 case. To date, this is the only case where detailed results are knownfor arbitrary couplings and eigenvalues. As such, it can serve as a source ofexpectations/conjectures for the elliptic and general-N cases, to which wereturn shortly.

Subsection 5.3 starts from a quite special type of eigenfunction (‘groundstate’) that arises for arbitrary N , both on the hyperbolic and ellipticlevels. Its existence is an immediate consequence of the above lemmas, andaccordingly we focus on non-obvious consequences, in particular as regardsparameter symmetries and its role in eventual Hilbert space reformulationsof the pertinent AΔO actions.

We continue with a general appraisal of the latter context. First, werecall that even when one is dealing with a specific AΔO A that is formallyself-adjoint on a specific Hilbert space H, it need not be true that A canbe naturally associated with a bona fide self-adjoint operator A acting on adense subspace of H. (Since AΔOs are unbounded, A cannot act on all ofH.) Furthermore, the association need not be unique: even for free AΔOsone can easily arrive at an infinity of distinct Hilbert space operators A.(See [24, 23] for explicit examples.) Here and below, we are using a dotfor Hilbert space operators, since the more obvious hat would give rise toconfusion with dual objects.

In all of the concrete cases known to us, the map

A $→ A, (5.1)


from AΔOs to self-adjoint operators is defined by using explicit A-eigenfunctionswith very special features. The ambiguity in the map reflects the hugeambiguity in A-eigenfunctions, once their existence has been established.

In the special cases at issue in this paper, however, the ambiguity canbe greatly reduced by extra requirements. Consider first the N = 1 setting,as detailed in Section 3. Suppose we restrict attention to eigenfunctionsthat are real-analytic and symmetric in a+ and a−, cf. our discussion inthe paragraph containing (4.32). There is considerable evidence (but nocomplete proof) that there exists at most one meromorphic eigenfunctionwith this property (up to multiplication by (a+ ↔ a−)-symmetric con-stants, of course). For the hyperbolic case, we detail such an eigenfunctionin Subsection 5.2. For the elliptic case no general coupling eigenfunctionsare known at all, but in a special case with only one variable coupling,there are partial results that are consistent with the required symmetry ina+, a− [25].

The BC1 eigenfunctions just mentioned are also invariant under theremaining parameter symmetries discussed above. More precisely, the H-and A-eigenfunctions are D4-invariant in the hyperbolic case [16], and ZZ2-invariant with respect to the single coupling in the elliptic case [25]. Weshould stress that these invariances do not necessarily follow from the AΔOsbeing invariant.

Turning to the arbitrary-N case, there is an obvious extra requirement(in addition to BCN and parameter invariances): one should search forjoint eigenfunctions of the N commuting AΔOs. This is the first point inthis paper where the latter are explicitly mentioned, and indeed, we havenot shown that they share the parameter symmetries of the defining AΔO.Even so, it is quite plausible that this is true, and we also believe there ex-ist BCN -invariant joint eigenfunctions that generalize our BC1 hyperboliceigenfunction, with the parameter symmetries discussed above.

To date, there is admittedly precious little evidence for the existence ofjoint eigenfunctions with all of these features. Koornwinder’s BCN polyno-mials [11] should arise as discretizations of the hyperbolic BCN joint eigen-function, just as the Askey-Wilson polynomials arise by discretizing ourBC1 eigenfunction, cf. Subsection 5.2. But these polynomial discretizationsonly yield fragmentary tests of the expected scenario.

Considerably stronger results that are relevant in this connection wererecently obtained by Chalykh [26]. He presents eigenfunctions for the hyper-bolic BCN case, which interpolate the Koornwinder polynomials. However,his arbitrary-eigenvalue elementary eigenfunctions are tied to a lattice ofcoupling constants, which should in turn be interpolated. Even so, inasmuchas his eigenfunctions allow tests of the above scenario, these tests seem tobe passed.


Chalykh’s results also exhibit a further symmetry feature that is not atall visible from the defining hyperbolic AΔO. (This is why we have not hadoccasion to mention this property in previous sections.) Roughly speak-ing, his eigenfunctions are symmetric under interchange of the variablesx1, . . . , xN occurring in the AΔO and suitable spectral variables p1, . . . , pN .Therefore, they are also eigenfunctions of an AΔO of the same structureas the defining one, which depends on the variables p1, . . . , pN .

We will make this self-duality property more precise in Subsection 5.2.Again, one may expect that it will persist for the N > 1 and arbitrary-coupling hyperbolic eigenfunctions that are yet to be found/invented, in ac-cordance with self-duality properties of the arbitrary-coupling Koornwinderpolynomials [27].

5.2. THE HYPERBOLIC BC1 CASE

Although the AΔOs H and A from Subsection 3.1 admit a manifestly D4-invariant representation, this does not imply that a given eigenfunction hasthis property. The eigenfunctions at issue in this subsection do have D4-invariance, but as will become apparent shortly, this symmetry property isfar from manifest.

More precisely, we are not aware of any representation for which D4-invariance is clear by inspection. The representation we are about to detailis the only one known to date, and one of its key features is the specialcharacter of one parameter. (Permutation invariance with respect to thethree remaining parameters is manifest.)

Turning to the details, we are going to introduce an A-eigenfunctionR(a+, a−, c; z, z) with eigenvalue 2c+(2z), z ∈ C| . Here, c ∈ C| 4 is thecoupling vector that is related to the parameter vector h by (3.21). It isexpedient to introduce dual couplings and parameters by setting

c ≡ Jc, h ≡ Jh, J ≡ 1

2

⎛⎜⎜⎝1 1 1 11 1 −1 −11 −1 1 −11 −1 −1 1

⎞⎟⎟⎠ . (5.2)

Since the shift vector (a, a−/2, a+/2, 0) in (3.21) is J-invariant, h and c areonce more related via (3.21).

To define the R-function, we also need quantities

s1 ≡ c0 + c1 − a−/2, s2 ≡ c0 + c2 − a+/2, s3 ≡ c0 + c3, (5.3)

and functions

F (b; y, z) ≡ G(z + y + ib− ia)G(y + ib− ia)

G(z − y + ib− ia)G(−y + ib− ia) , (5.4)


K(c; z) ≡ 1

G(z + ia)

3∏j=1

G(isj)

G(z + isj), (5.5)

where G(a+, a−; z) is the hyperbolic gamma function, cf. Section 2. Choos-ing now at first

c ∈ IR4, Re z,Re z > 0, s1, s2, s3 ∈ (−a, a), (5.6)

the R-function is defined by the contour integral

R(c; z, z) =1

(a+a−)1/2

∫CF (c0; z, z

′)K(c; z′)F (c0; z, z′)dz′. (5.7)

The choice of the contour C is determined by the location of the polesin the integrand. The function K(c; z′) gives rise to four upward pole se-quences on the imaginary axis, beginning at z′ = 0, i(a − sj), j = 1, 2, 3,whereas F (b; y, z′) yields two downward sequences, beginning at z′ = ±y−ib. The contour is given by a horizontal line Im z′ = d, indented (if neces-sary) so that it passes above the points −z − ic0,−z − ic0 in the left halfplane and z− ic0, z− ic0 in the right half plane, and so that it passes below0. Thus the four upward pole sequences of the integrand are above C andthe four downward ones are below C. The integrand has exponential decayas |Re z′| → ∞, so that the integral does not depend on d.

Since we have

isj = h0 + hj + ia, j = 1, 2, 3, ic0 =1

2

3∑n=0

hn + ia, (5.8)

the function R(a+, a−, c(h); z, z) is manifestly invariant under permuta-tions of h1, h2, h3, whereas h0 plays a very different role. Viewed as afunction of a+, a− and h, it is moreover manifestly invariant under a+ ↔a−. Taking for granted that it satisfies the announced eigenvalue AΔE

AR(a+, a−, c(h); z, z) = 2c+(2z)R(a+, a−, c(h); z, z), (5.9)

it is therefore immediate that we also have

AR(a+, a−, c(h); z, z) = 2c−(2z)R(a+, a−, c(h); z, z), (5.10)

where A is obtained from A by interchanging a+ and a−. Furthermore,noting

isj = h0 + hj + ia, j = 1, 2, 3, (5.11)

it follows that R is an eigenfunction with eigenvalues 2c±(2z) of two AΔOsacting on z; the latter are obtained from A and A by taking h → h andz → z. (See [15] for the details of the results mentioned thus far.)


To retain the latter self-duality property, the similarity transformationleading from A to A (cf. (3.7)) should be performed on the dual variable zas well. Moreover, a different normalization turns out to be expedient forthe A-eigenfunction. Specifically, following [16] we work with the function

E(h; z, z) ≡ χ(h)[c(h; z)c(h; z)3∏j=1

G(isj)]−1R(c(h); z, z), (5.12)

where c(h; z) is defined by (3.1) and χ is the phase

χ(h) ≡ exp(−iα[3∑n=0

h2n/4 + (a2+ + a

2− + a+a−)/8]). (5.13)

With these conventions, the E-function has plane wave asymptoticsE(h; z, z) ∼ exp(iαzz)− u(h;−z) exp(−iαzz), z →∞, (5.14)

where the u-function is given by (3.4) (with r = 0, of course). Moreover,denoting the D4 Weyl group by W , we have

E(w(h); z, z) = E(h; z, z), ∀w ∈W. (5.15)

In words, the E-function is D4-invariant.The latter invariance property and (a far stronger version of) the asymp-

totics (5.14) are the principal results of [16]. Even though these two resultsare quite distinct at face value, their proofs in [16] are intimately related.Briefly, the asymptotics is obtained first for a restricted parameter re-gion that is W -invariant. Next, the W -invariance of the asymptotics isused to prove W -invariance of E for this region. By analytic continuation,W -invariance for arbitrary parameters then follows. The asymptotics forarbitrary parameters can now be established by exploiting W -invariance.

As we have already seen at the end of Subsection 3.1, we need to choosethe couplings real to obtain formally self-adjoint AΔOs. In [17] we havestudied the question whether these formal insights can be improved toa rigorous Hilbert space formulation. In view of the meromorphy, planewave asymptotics, and D4-invariance of the E-function, this problem isbest studied by viewing E(a+, a−,h; z, z) (for ih ∈ IR4) as the kernel of amap between Hilbert spaces

H ≡ L2((0,∞), dz), H ≡ L2((0,∞), dz). (5.16)

More specifically, our starting point in [17] is the linear operator

F : C∞0 ((0,∞)) ⊂ H → H,

φ(z) $→(α

2π

)1/2 ∫ ∞

0E(z, z)φ(z)dz. (5.17)


This eigenfunction transform (generalized Fourier transformation) is welldefined for generic parameters in iIR4. (The integral converges for all h ∈iIR4 and z > 0, but for special parameters the E-function may have a poleat z = 0, precluding an image in H.)

In spite of the normalized plane wave asymptotics (5.14) (which is for-mally unitary) and the formal self-adjointness of A on H, the operator Fis not isometric in general, and (accordingly) the operator A defined onFC∞0 ((0,∞)) by setting

AF = FM, (5.18)

where M is the operator of multiplication by 2c+(2z), is not symmetric(hermitian) in general. But when the parameters (a+, a−, ih) are requiredto belong to the polytope

P ≡ {(a+, a−, p) ∈ (0,∞)2 × IR4 | |pn| < a, n = 0, . . . , 3}, (5.19)

then F is isometric and (hence) A is symmetric.

Requiring not only (a+, a−, ih) ∈ P , but also (a+, a−, ih) ∈ P , theoperator F extends to an isometry from H onto H, and A extends to aself-adjoint operator on (a certain dense subspace of) H. As a consequence,for this parameter set we then obtain D4-invariant operators F and A.

The case where one has (a+, a−, ih) ∈ P , but (a+, a−, ih) /∈ P , is farricher, since bound states appear on the stage. We proceed by taking acloser look at this situation. As a bonus, this prepares the ground for thenext subsection.

To begin with, the bound states take their simplest form in terms of theR-function (as compared to the E-function). Specifically, assuming fromnow on a− ≤ a+, they arise from the discrete z-choices ic0 + ina−, n ∈ IN ,where we have

R(a+, a−, c; z, ic0 + ina−) = Pn(c+(2z)), n ∈ IN, (5.20)

for generic parameters. Here, Pn(u) is a degree-n polynomial in u; thesepolynomials are basically the Askey-Wilson polynomials [12, 13].

In particular, for n = 0 we obtain the constant function

P0(u) = 1, (5.21)

and the remaining polynomials can be obtained via the AΔO dual to A(it yields the three-term recurrence). Even though these eigenfunctions aresingularity-free for generic parameters, they do not enter the above Hilbertspace formulation whenever we have

(a+, a−, ih) ∈ P, (a+, a−, ih) ∈ P. (5.22)


Indeed, as already mentioned, the eigenfunction transform F is unitary inthis case, and its kernel consists of ‘scattering states’, cf. (5.17), (5.14).

This critical dependence on the parameters can already be appreciatedby focusing on the ‘ground state’ (5.21). To start with, note that by (3.7)and (3.17) we have

A = V (z)[exp(−ia−∂z)− 1]+ V (−z)[exp(ia−∂z)− 1]− 2c+(

∑hn + ia−). (5.23)

It is therefore immediate that the constant functions are A-eigenfunctionswith eigenvalue −2c+(

∑hn + ia−). Recalling (3.21), one sees that this is

consistent with what we asserted in (5.20)–(5.21), cf. (5.9). (Note howeverthat a priori the R-function might be a quite different eigenfunction forz = ic0.)

Let us now pass to the transformed eigenvalue equation

A(1/c(h; z)) = −2c+(∑hn + ia−)/c(h; z). (5.24)

The first important point is that the A-eigenfunction 1/c(h; z) does notbelong to H for parameters satisfying (5.22). To be sure, this is not clear byinspection. But the known asymptotics of the hyperbolic gamma functionfor |Re z| → ∞ can be invoked to obtain (cf. (3.5) in [16])

c(h; z)−1 ∼ χ(h)−1 exp(αz(a− ih0)), z →∞. (5.25)

Accepting this, it is clear by inspection that the eigenfunction divergesexponentially for ih0 < a.

As a second key point, (5.25) entails that 1/c(h; z) is indeed square-

integrable at ∞ whenever ih0 > a. Now for parameters in P one readilychecks that ih0 can vary over all of (−2a, 2a). Furthermore, for ih0 ∈ (a, 2a)the function 1/c(h; z) is not only in H, but also orthogonal to the range ofF . Defining the action of A on the H-vector

ψ0(z) ≡ 1/c(h; z), (a+, a−, ih) ∈ P, ih0 ∈ (a, 2a), (5.26)

to be equal to that of A, we obtain therefore a bound state for A (namely,its ground state), orthogonal to the ‘scattering states’ in the range of F .

Depending on the precise relation between ih0 and the parameters a+and a−, there are further bound states Pn(c+(2z))/c(h; z). In view of theasymptotics (5.25), it is actually easy to see for which degrees n ∈ INone obtains exponential decay for z → ∞, hence square-integrability: oneclearly needs

na− + a− ih0 < 0. (5.27)


For far more information on these bound states we refer to [17]; our re-maining discussion in this subsection concerns the issue of D4-invariancevs. the ground state (5.26).

We begin by noting that the requirement

i(h0 + h1 + h2 + h3) ∈ (2a, 4a) (5.28)

is S4-invariant, but when we flip an even number of signs of ih0, . . . , ih3,then it is no longer satisfied. At first sight, this seems to be in conflict withthe D4-invariance we emphasized above. But in fact there is no contradic-tion here. Indeed, since A is D4-invariant, we conclude from (5.24) that forall w ∈W the function 1/c(w(h); z) is an A-eigenfunction with eigenvalue−2c+(

∑w(h)n+ia−). Since there are 8 distinct even sign flips, we obtain 8

distinct eigenfunctions with 8 distinct eigenvalues (for generic parameters).Whenever (a+, a−, ih) ∈ P satisfies

η ≡ max(|h0|, |h1|, |h2|, |h3|) > a, (5.29)

there is a unique n such that η = |hn|, cf. (5.2). A suitable W -transform-ation then ensures

i(Jw(h))0 ∈ (a, 2a), (5.30)

and so the above R-function properties hold true for w(h).Therefore, a W -invariant definition of the Hilbert space ground state

reads as follows. First, we require (a+, a−, ih) to belong to the (W -invariant)polytope P (5.19). Provided the parameters also satisfy the (W -invariant)restriction (5.29), we choose w ∈W such that (5.30) holds true, and definethe ground state by

ψ0(z) ≡ 1/c(w(h); z). (5.31)

Then we obtain the W -invariant eigenvalue equation

Aψ0(z) = 2 cos(2π(η − a)/a+)ψ0(z), η ∈ (a, 2a). (5.32)

Switching to theW -invariant AΔOH(a+, a−,h; z) with (a+, a−,h) ∈ P(cf. (3.10)–(3.13)), we see from the above discussion that the positive evenfunction w(h; z)1/2, z ∈ IR, should only be viewed as a ground state whenit is square-integrable over IR. Among the 8 (generically) distinct functionsw(w(h); z), w ∈W , there is at most one with this property.

5.3. GROUND STATES

From Lemma 4.1 we obtain the arbitrary-N version of the A-representation(5.23), namely (cf. (4.14), (4.26))

A =N∑j=1

(Vj(x)[exp(−ia−∂j)−1]+Vj(−x)[exp(ia−∂j)−1])+E(h, μ), (5.33)


E(h, μ) ≡ −2s+(Nμ)s+(μ)

c+

(3∑n=0

hn + (N − 1)μ+ ia−). (5.34)

Just as forN = 1, it is plain that the constant functions areA-eigenfunctionswith eigenvalue E(h, μ). SinceH(h, μ) isD4-invariant in h and ZZ2-invariantin μ (recall the paragraph containing (4.27)), the functionW (w(h), σ(μ);x)1/2

is an H(h, μ)-eigenfunction with eigenvalue E(w(h), σ(μ)) for all (w, σ) ∈W × ZZ2. Thus we obtain 16 (generically) distinct eigenfunctions.

Requiring from now on ih ∈ IR4, iμ ∈ IR, the functionsW (w(h), σ(μ);x)are positive (recall (4.1)–(4.5)). For μ = 0 we obtain a product of BC1

weight functions, and we can use the results of the previous subsection.Thus, choosing (a+, a−, ih) ∈ P , we need ih0 ∈ (a, 2a) forW (a+, a−,h, 0;x)to have a finite integral over the BCN Weyl chamber GN (4.29), whereasthe 7 other W -transforms are not integrable over GN . Clearly, this remainstrue when we let μ = iτ with τ ∈ (−ε, ε) and ε ∈ (0, a) small enough; forthe transformed τ -values 2a− τ the W -integral over GN diverges, however.

More generally, it is probably true that among the 16 distinct weightfunctions related byW×ZZ2 transformations, at most one is integrable overGN . In the absence of eigenfunctions for arbitrary N , arbitrary parametersand arbitrary eigenvalues, it is however not clear under what restrictionson the parameters the function W (h, μ;x)1/2 serves as the ground stateof a self-adjoint operator H on L2(GN , dx), whose action coincides withthat of the AΔO H on a dense subspace. (Note that due to the similaritiesconnecting H with A and A, this is equivalent to 1/C(h, μ;x) being theground state of A on L2(GN , dx), and the constant function being theground state of A on L2(GN ,W (x)dx).)

Turning to the elliptic case, we need to impose the restriction (4.33) on(h, μ) for the AΔO A to take the form (5.33), with

E(h, μ) =N∑j=1

(Vj(h, μ;x) + Vj(h, μ;−x)) + V(h, μ;x), (5.35)

cf. Lemma 4.2. Doing so from now on, it is once again obvious that theconstant function is an A-eigenfunction with eigenvalue E(h, μ). HenceW (h, μ;x)1/2 is an H(h, μ)-eigenfunction with eigenvalue E(h, μ).

Below Lemma 4.2 we have already seen one way to obtain a positiveweight function, namely via real couplings c and τ . But this is not theonly possibility: we can also add iπ/2r to each of these couplings. For all ofthese choices, the functionW (h, μ;x) is integrable over the Weyl alcove GN(4.34). (As before, this is true for generic parameters—taking for exampleN = 1, one can choose special parameters so that w(h; z) has a pole ofarbitrary order at z = 0, precluding integrability.) Moreover, choices forwhich (4.33) holds true yield H(h, μ)-eigenfunctions W (h, μ;x)1/2.


Admittedly, in the elliptic case it is not even clear for N = 1 which ofthe above positive H-eigenfunctions admit a reinterpretation as the groundstate of a self-adjoint Hilbert space operator H. A natural expectation ishowever that the choice of (h, μ) should be such that the hyperbolic limitnot only exists, but also yields a weight function that is integrable overIRN . In this connection we point out that the limit r ↓ 0 need not exist forarbitrary shifts of the couplings by iπ/2r. For instance, taking ih ∈ IR8,the N = 1 weight function w(r, a+, a−,h; z) has no r ↓ 0 limit.

Fixing c ∈ IR8 and τ ∈ IR, it is easily verified that the limit does exist,yielding the hyperbolic weight function corresponding to (c0, c1, c2, c3) andτ . Indeed, using the limits (2.15) and (2.16), one deduces that the r ↓ 0limit exists already for the c-functions (3.1) and (4.1).

6. Nonrelativistic limits

In this section we complete our account by studying various aspects of thenonrelativistic limit. More specifically, we obtain the limits of the aboveAΔOs and eigenfunctions, paying special attention to their parameter sym-metries.

To begin with, the symmetry under interchange of a+ and a− (whichis evidently present for the eigenfunctions 1,W (x)1/2, 1/C(x) of the AΔOsA,H and A, resp.) disappears, since the limit involves sending one of a+, a−to 0. In the hyperbolic N = 1 case, the relevant limit is defined by setting

c = a−(d0, . . . , d3), (6.1)

(with c related to h via (3.21)), and taking a− to 0 for fixed d. It isconvenient to introduce new parameters

ν = π/a+, g02 = d0 + d2, g13 = d1 + d3. (6.2)

For the functions V, Va and Vb we now calculate the expansions

F (z) = F (0) + a−F (1)(z) + a2−F(2)(z) +O(a3−), a− ↓ 0,

F = V, Va, Vb. (6.3)

The results read

V (0) = 1, V (1)(z) = −iν(g02 coth(νz) + g13 tanh(νz)), (6.4)

2V (2)(z) = −ν2(2d0d2 − d2sinh2(νz)

− 2d1d3 − d3cosh2(νz)

+ (g02 + g13)2

), (6.5)

V (0)a = 1, V (1)

a = 0, (6.6)


V (2)a (z) = ν2

(d20 + d

22 − d0

sinh2(νz)− d

21 + d

23 − d1

cosh2(νz)

), (6.7)

V(0)b = 0, V

(1)b = 0, (6.8)

V(2)b (z) = ν2

(2d0d2 − d2sinh2(νz)

− 2d1d3 − d3cosh2(νz)

). (6.9)

For the AΔOs A,H and A we then obtain

D = 2 + a2−Dnr +O(a3−), a− ↓ 0, D = A,H,A, (6.10)

where

Anr = −d2

dz2− 2iV (1)(z)

d

dz− ν2(g02 + g13)2, (6.11)

Hnr,Anr = −d2

dz2+ Vh(z), (6.12)

with the hyperbolic potential

Vh(z) =ν2g02(g02 − 1)sinh2(νz)

− ν2g13(g13 − 1)cosh2(νz)

. (6.13)

Next, we deduce from the limit (2.7) that we have

lima−↓0

w(a+, a−,h(c); z) = [2 sinh(νz)]2g02 [2 cosh(νz)]2g13 ≡ wnr(z),

z ∈ (0,∞). (6.14)

As a check, note that we do get the expected relation

Hnr = wnr(z)1/2Anrwnr(z)

−1/2, (6.15)

cf. (3.13). (The equality of Hnr and Anr can be understood from unr beingz-independent.)

The obvious expectation is that the manifest symmetries under g02 →1−g02 and g13 → 1−g13 of Hnr are remnants of the hidden D4-symmetriesof A and H. In point of fact, this is only true for the second involution;the first one is a new symmetry. To appreciate this, one need only rewrite(6.1) in terms of h via (3.21) and reinspect the limiting behavior: the g13 →1 − g13 map arises from the sign flip (h1, h3) → (−h1,−h3), whereas theg02 → 1− g02 map has no h-analog.

This different ancestry of the two involutions can also be recognizedfrom their different effect on the nonrelativistic limit of the D4-invariantBC1 eigenfunction E(a+, a−,h; z, z): the limit function is invariant underg13 → 1− g13, but not under g02 → 1− g02.


To substantiate this assertion, we recall from [15] that for the R-functionto converge to 2F1 we should not only rescale the couplings by a− (cf. (6.1)),but also the dual variable z. Here it is convenient to set

z = a−k, (6.16)

obtaining [15]

lima−↓0

R(a+, a−, a−d; z, a−k) = 2F1(α, β, γ;− sinh2(νz)),

z ∈ (0,∞), k ∈ IR, (6.17)

with

α = (g02 + g13)/2− ik, β = (g02 + g13)/2 + ik, γ = g02 + 1/2. (6.18)

Using (2.7), we calculate

lima−↓0

χ(h)/c(h; z) = wnr(z)1/2, z ∈ (0,∞), (6.19)

and using also (2.8), we get

lima−↓0

c(h; a−k)3∏j=1

G(isj) = cnr(k)Γ(g02 + 1/2), (6.20)

cnr(k) ≡ Γ(ik)Γ(ik + 1/2)/Γ(ik + (g02 + g13)/2)Γ(ik + (g02 + 1− g13)/2).(6.21)

Recalling (5.12), we can put the pieces together, obtaining

lima−↓0E(z, a−k) = wnr(z)

1/2cnr(k)−1

2F1(α, β, γ;− sinh2(νz)),

z ∈ (0,∞), k ∈ IR. (6.22)

Now cnr(k) is obviously invariant under g13 → 1− g13. The asserted invari-ance of the E-limit (6.22) therefore amounts to the identity

2F1(α, β, γ;x) = (1− x)γ−α−β2F1(γ − β, γ − α, γ;x), (6.23)

which was already known to Gauss (see e.g. 9.131 in [28]).As announced, the limit (6.22) is not invariant under g02 → 1− g02. To

be sure, the transformed function is also an eigenfunction of the differentialoperator Hnr. For g02 ∈ (−1/2, 3/2) and g13 ∈ IR, these two eigenfunctionsare associated to distinct self-adjoint operators H±nr on L2((0,∞), dz), withH+

nr defined for g02 ∈ [1/2,∞) and H−nr for g02 ∈ (−1/2, 1/2]. (Of course,for g02 = 1/2 the two operators coincide.)


For the arbitrary-N hyperbolic case we should also rescale the couplingτ by a−. That is, we should set

τ = a−g, (6.24)

before taking a− to 0. Since the parameter symmetry τ → 2a−τ (cf. (4.27))amounts to a map g → 1 − g + a+/a− that has no limit for a− ↓ 0, thissymmetry does not persist on the nonrelativistic level. But a new invarianceunder g → 1 − g appears, as will be clear from the formulas that follow.This is a manifest invariance property of Hnr, but in cases where this canbe checked, the analytic continuation from g ∈ [1/2,∞) to g < 1/2 of theeigenfunction that is relevant for Hilbert space purposes is not invariant. (Asbefore, it is associated to a distinct self-adjoint extension of the differentialoperator Hnr restricted to smooth functions with compact support in GN .)

To expand the coefficients of the AΔOs, we can use our previous calcu-lations (6.3)–(6.9), not only for the external field terms V (±xj), but alsofor the extra μ-dependent terms. (This is because they give rise to functionsof the form V (xj ± xk), Va(xj ± xk), with d0 = g, dj = 0, j > 0.) Moreover,it is not hard to check

V(x) = a2−N∑j=1

V(2)b (xj) +O(a

3−), a− ↓ 0. (6.25)

The result is that we obtain expansions

D = 2N + a2−Dnr +O(a3−), a− ↓ 0, D = A,H,A, (6.26)

where

Anr =N∑j=1

(−∂2j − 2

(iV (1)(xj) (6.27)

+ νg∑k �=j

[coth ν(xj − xk) + coth ν(xj + xk)])∂j)+ E0,

E0 = −Nν2((g02 + g13)

2 + 2(N − 1)g(g02 + g13) (6.28)

+2

3(2N − 1)(N − 1)g2

),

Anr,Hnr =N∑j=1

(−∂2j + Vh(xj) (6.29)

+∑k �=j

( ν2g(g − 1)sinh2 ν(xj − xk)

+ν2g(g − 1)

sinh2 ν(xj + xk)

)),


with V (1)(z) and Vh(z) given by (6.4) and (6.13), resp. We used (5.33)–(5.34) to simplify the expansion of A.

From the limit (2.7) we deduce

lima−↓0

W (x) =N∏j=1

wnr(xj) ·∏

1≤j<k≤N[4 sinh ν(xj − xk) sinh ν(xj + xk)]2g

≡ Wnr(x), x ∈ GN . (6.30)

The constant function is manifestly an Anr-eigenfunction with eigenvalueE0, but the ‘ground state’ equation

HnrWnr(x)1/2 = E0Wnr(x)

1/2 (6.31)

is not obvious. It amounts to a functional equation that can be obtainedfrom the a− ↓ 0 limit of Lemma 4.1. The latter also yields a direct proof ofthe similarity

Anr =Wnr(x)−1/2HnrWnr(x)

1/2, (6.32)

which is the limit of (4.7).Next, we consider the elliptic N = 1 case. As in the hyperbolic case, we

should first switch from the parameter vector h to the coupling vector c(now given by (3.21) and (3.43)), and set

c = a−(d0, . . . , d7). (6.33)

Taking a− to 0 for fixed d, we can obtain the pertinent expansions via(2.23) and (2.25) (with a→ a+). It is convenient to introduce

g0 = d0 + d2, g1 = d4 + d6, g2 = d1 + d3, g3 = d5 + d7, γ = r(g0 + g1),(6.34)

the shifted ℘-function℘(z) = ℘(z) + η/ω1, (6.35)

and the sum of logarithmic derivatives

L+(z) = g0s′+(z)s+(z)

+ g1s′+(z + ω1)s+(z + ω1)

+ g2R′+(z)R+(z)

+ g3R′+(z + ω1)R+(z + ω1)

. (6.36)

Now we obtain by long, but straightforward calculations the elliptic gener-alization of (6.4)–(6.9):

V (0) = 1, V (1)(z) = −iL+(z)− γ, (6.37)

2V (2)(z) = V (1)(z)2 + (d20 + d22 + d2)℘(z) + (d

24 + d

26 + d6)℘(z + ω1)

+ (d21 + d23 + d3)℘(z + ω2)

+ (d25 + d27 + d7)℘(z + ω3), (6.38)


V (0)a = 1, V (1)

a = −2γ, (6.39)

V (2)a (z) = 2γ2 + (d20 + d

22 − d0)℘(z) + (d24 + d26 − d4)℘(z + ω1)

+ (d21 + d23 − d1)℘(z + ω2)

+ (d25 + d27 − d5)℘(z + ω3), (6.40)

V(0)b = 0, V

(1)b = 0, (6.41)

V(2)b (z) = (2d0d2 − d2)℘(z) + (2d4d6 − d6)℘(z + ω1)

+ (2d1d3 − d3)℘(z + ω2)+ (2d5d7 − d7)℘(z + ω3) + cb, (6.42)

cb = −e1(2d0d2 − d2 + 2d4d6 − d6)− e3(2d1d3 − d3 + 2d5d7 − d7),ej = ℘(ωj). (6.43)

Therefore, the AΔOs have an expansion of the form

D = 2− 2a−γ + a2−Dnr +O(a3−), a− ↓ 0, D = A,H,A, (6.44)

where we have

Anr = −d2

dz2− 2L+(z)

d

dz− L+(z)2 +

3∑t=0

g2t ℘(z + ωt) + χA, (6.45)

Hnr = −d2

dz2+ Ve(z) + χH , (6.46)

Anr = −d2

dz2− 2iγ d

dz+ Ve(z) + χA, (6.47)

with the elliptic potential

Ve(z) =3∑t=0

gt(gt − 1)℘(z + ωt), (6.48)

and the constants

χH = γ2 + cb +

(7∑n=0

d2n − d0 − d1 − d4 − d5)η/ω1, (6.49)

χA = χH + γ2, (6.50)

χA = χH + 2

(7∑n=0

dn

)η/ω1. (6.51)


From the limit (2.17) we see that the c-, u- and w-functions have limitsfor z ∈ (0, ω1) given by

cnr(z) = R+(z + ω2)−g0R+(z − ω3)−g1R+(z)

−g2R+(z + ω1)−g3 , (6.52)

unr(z) = exp((g0 + g1 − 1)(2irz − iπ)), (6.53)

wnr(z) = pws+(z)2g0s+(z + ω1)

2g1R+(z)2g2R+(z + ω1)

2g3 , (6.54)

with pw a positive constant. Once again, (6.15) can be checked directly, justas the relation

Hnr = (−e2irzunr(z))1/2Anr(−e2irzunr(z))−1/2. (6.55)

Rewriting (6.33) in terms of h, we see that only the sign flips on h1, h3and on h5, h7 survive in the limit. (Indeed, Hnr and Anr are invariant underd1, d3 → 1 − d1,−d3 and under d5, d7 → 1 − d5,−d7.) When we omit theconstants in (6.46) and (6.47), these involutions amount to g2 → 1 − g2and g3 → 1 − g3, cf. (6.34). But the invariances under g0 → 1 − g0 andg1 → 1 − g1 are new. Since general coupling eigenfunctions are neitherknown for the relativistic nor for the nonrelativistic case, we cannot studythese parameter symmetries any further.

As we have seen above, the relation (3.45) guarantees that w(h(c); z)1/2

is an H-eigenfunction. In terms of d (6.33), this says∑dj = 0. Therefore,

requiring3∑t=0

gt = 0, (6.56)

we deduce that the functions wnr(z)1/2, 1/cnr(z) and the constant function

are eigenfunctions of Hnr,Anr and Anr, resp. This is by no means clear byinspection, but can also be checked directly, as follows.

First, the condition (6.56) ensures that L+(z) (6.36) is elliptic. Second,it clearly implies

L′+(z) = −3∑t=0

gt℘(z + ωt),(∑

gt = 0). (6.57)

Third, L+(z) is odd, so L+(z)2 is even. Comparing principal parts at poles,one deduces that L+(z)2 −

∑g2t ℘(z + ωt) is constant:

L+(z)2 =∑t

g2t ℘(z + ωt) + C(g),(∑

gt = 0). (6.58)

A moment’s thought now reveals that (6.57)–(6.58) entail the relevanteigenfunction properties.


Next, we consider the arbitrary-N elliptic case. Proceeding as in thehyperbolic case, we note that with the reparametrization (6.24) in effect,the N = 1 calculations (6.37)–(6.43) can also be applied to the μ-dependentterms. Furthermore, the expansion (6.25) is still valid on the elliptic level.As a result, we obtain expansions of the form

D = 2N − 2Na−ζ + a2−Dnr +O(a3−), a− ↓ 0, D = A,H,A, (6.59)

whereζ = γ + 2(N − 1)rg. (6.60)

In particular, we getHnr = 2H(BC)

nr + χ, (6.61)

withH(BC)nr the Inozemtsev Hamiltonian (1.3) and χ a constant. For brevity,

we omit the explicit form of χ and of Anr and Anr.From the limit (2.17) we deduce

lima−↓0

W (x) = pN∏j=1

wnr(xj) ·∏

1≤j<k≤N[s+(xj − xk)s+(xj + xk)]2g

≡ Wnr(x), x ∈ GN , (6.62)

with p a positive constant, wnr(z) given by (6.54), and GN by (4.34). Theexplicit form of the differential operator Anr in (6.59) can also be calculatedby using (6.61), (6.62) and (6.32).

The constraint (4.33) turns into

3∑t=0

gt + 2(N − 1)g = 0. (6.63)

With this restriction of the couplings, we obtain a factorized ‘ground state’eigenfunction Wnr(x)

1/2 of the Inozemtsev Hamiltonian (1.3). The associ-ated functional equation follows from the a− ↓ 0 limit of Lemma 4.2.

We proceed with a remark about the expansions (6.59). Suppose wework instead of D with the renormalized AΔO

Dren = ρrenD, ρren = exp[r(c0 + c2 + c4 + c6 + 2(N − 1)τ)]. (6.64)

The multiplier ρren destroys the D8-invariance of H and A, but it entailsthe simpler expansions

Dren = 2N + a2−(Dnr −Nζ2) +O(a3−), a− ↓ 0, D = A,H,A. (6.65)

This observation is also relevant for the relation between our AΔO H (4.6)and the AΔO HD introduced by Hikami and Komori [9]. Indeed, as we


detail in Appendix B, these two operators are connected by a multiplicativeconstant that amounts to ρren, cf. (B.15)–(B.16).

We conclude this section with additional comments on related litera-ture. First, we note that Hikami and Komori [9] connected their versionsof the elliptic AΔOs A and H by a similarity transformation that doesnot amount to our manifestly (a+ ↔ a−)-invariant transformation (4.7)involving elliptic gamma functions. The difference consists in multipliersthat are elliptic with periods π/r and ia−, so that there is no contradictionwith our similarity transformation. But their elliptic multipliers do not havea limit for a− ↓ 0, so that their different weight function has no limit either.

Another item of special note is the connection of the nonrelativisticlimits of the above (candidates for) ground states to a recent series ofpapers by Gomez-Ullate et al., cf. in particular [18, 19]. They are focusingon more general Calogero-Moser type systems, since their work pertains toparticles with internal degrees of freedom. Even so, their results do admit aspecialization to the context of this section, since one can restrict attentionto ‘scalar’ wave functions with suitable symmetries. Doing so, one obtains inparticular finite-dimensional invariant spaces for the Inozemtsev operator(1.3) (hence for Hnr (6.61)), provided the couplings are suitably restricted.In case the invariant subspace is one-dimensional, the restriction amountsto our restriction (6.63). At face value, the positive Hnr-eigenfunctionsthey obtain for this case look quite different from our Hnr-eigenfunctionsWnr(x)

1/2. But in fact it can be verified that one is dealing with distinctrepresentations for the same functions.

Quite recently, the connection of the Spanish work to the Inozemt-sev Hamiltonian was also pointed out by Takemura [29]. He shows moregenerally that the finite-dimensional invariant subspaces admit a bijectivecorrespondence with subspaces of multi-dimensional theta functions thatare invariant under the AΔO H (4.6). The existence of the latter subspaces(for restricted parameters) was previously shown by Komori [30], cf. alsothe earlier work by Hasegawa and collaborators [31–33]. Probably, the moregeneral limiting transitions arising from Takemura’s paper [29] can againbe controlled by a suitable use of the elliptic gamma function.

Appendix A. Proofs of Lemmas 3.1, 3.2, 4.1 and 4.2

The four lemmas we prove in this appendix are closely related. Indeed,the N = 1 lemmas in Section 3 are partially implied by their arbitrary-N generalization in Section 4, and the hyperbolic lemmas can be partlyunderstood as the r ↓ 0 limit of their elliptic counterparts. Even so, it isilluminating to proceed ‘by generalization’, since we can obtain a little moreinformation in the special cases.


Proof of Lemma 3.1. Denoting the function on the rhs of (3.19) by F (z), wesee that Vb(z) (3.17) and F (z) are both even and ia+-periodic meromorphicfunctions with limit 0 for |Re z| → ∞, cf. (3.18). The poles in V (±z) atz = 0, ia+/2 do not occur in F (z), but their residues have opposite sign.To prove equality of F (z) and Vb(z), it therefore remains to verify equalityof residues at the (generically) simple poles z = ia and z = ia−/2. Using(3.15), this is routine. �

Proof of Lemma 3.2. The definitions (3.29) imply that the functions

Ft(μ; z) ≡Et(μ; z)− Et(μ; zt)

R+(μ− ia+/2)R+(μ− ia− − ia+/2), t = 0, . . . , 3, (A.1)

are not only elliptic in z, but also in μ. Calculating the residues for the(generically) simple poles at μ = 0 and μ = ia−, we see they vanish. Byellipticity, it follows that Ft(μ; z) has no poles for μ ≡ 0 and μ ≡ ia−, sothat it is constant in μ. Hence Vb(z) is μ-independent as well.

It remains to prove the last assertion. Due to (3.36), the function Σ(z)is elliptic with periods π/r, ia+. Since Σ(z) and R+(z) are even and π/r-periodic, the residues at the simple poles z ≡ 0, ω1 cancel. Using (2.18) and(3.36), we also verify that Σ(z) has no poles for z ≡ ω2, ω3.

Next, consider the residues at the simple poles z ≡ ia−/2+ωt of (3.39).Using (2.18), it is straightforward to check that they cancel. By evennessin z, no poles are present for z ≡ −ia−/2 − ωt either. Hence (3.39) doesnot depend on z. �

Proof of Lemma 4.1. The functions Σ(x) and V(x) are both ia+-periodicand even in x1, . . . , xN , and also SN -invariant. Comparing poles in x1 in aperiod strip, let us first note that the (generically simple) poles at x1 = ±xk,which occur in the summands of Σ(x), are not present in V(x). But it iseasily checked that residues at these poles cancel in pairs, so that the polesdo not occur in Σ(x) either. Thus we are left with the poles due to V (±x1)in V1(x) + V1(−x), and due to Qc(x1) and Qs(x1) in V(x).

More specifically, by evenness it suffices to consider the poles at x1 = iaand x1 = ia−/2. Just as for N = 1, it now follows from (3.15) that theresidues have opposite signs. Thus the function Σ(x)+V(x) has no poles inx1. Since it is ia+-periodic and has finite limits for |Re x1| → ∞, it does notdepend on x1. The first assertion now follows by permutation invariance.

To verify that the constant is given by (4.26), we need only take suc-cessively x1 →∞, . . . , xN →∞. For V(x) this yields limit 0, and for Σ(x)the constant on the rhs of (4.26). �


Proof of Lemma 4.2. Due to the restriction (4.33), not only V(x), but alsoΣ(x) is elliptic in x1, . . . , xN . Both functions are also even in x1, . . . , xN andpermutation invariant. As in the hyperbolic case, the poles at x1 ≡ ±xkin the terms of Σ(x) do not occur in Σ(x) due to residue cancellations. Toshow independence of x1, it is therefore enough to verify residue cancellationfor the x1-poles due to V (±x1) in V1(x) + V1(−x), and due to Et(x1), t =0, . . . , 3, in V(x).

To this purpose, the previousN = 1 calculations in the proof of Lemma 3.2can be used. To be specific, ‘unwanted’ 2ω2-shifts in the R+-argumentsarising in the products over k > 1 in V1(±x) can be removed via the R-AΔE (2.18). This yields extra terms (compared to N = 1) of the formexp(±4ir(N − 1)μ), which cancel due to the restriction (4.33). �

Appendix B. Comparison with Refs. [8,9]

The symmetry under interchange of a+ and a−, which is built into our defi-nitions, is not visible in the papers by van Diejen [8] and Komori/Hikami [9].To ease our notation and for comparison purposes, we abandon our sym-metric formulation in this appendix. Specifically, we change a+ to a andR+, s+, to R, s, and we follow van Diejen by using the parameter

γ ≡ ia−/2. (B.1)

(This parameter is denoted by β in [9].)As we shall now detail first, the analog of van Diejen’s external field

w(z) is our field V (−z), up to a certain factor. Indeed, w(z) is defined by(cf. Eq. (3.20) in [8])

w(z) =3∏t=0

σt(z + μt)σt(z + γ + μ′t)

σt(z)σt(z + γ). (B.2)

Using now successively (2.28), (2.25), (2.19) and (2.18), we obtain

w(z) = exp(CD +KDz)[R(2z − ω2)R(2z + 2γ − ω2)]−1×R(z + μ0 − ω2)R(z + μ2)R(z + μ′0 + γ − ω2)×R(z + μ′2 + γ)R(z + μ1 − ω1 − ω2)R(z + μ3 − ω1)×R(z + μ′1 + γ − ω1 − ω2)R(z + μ′3 + γ − ω1), (B.3)

CD ≡ η1rπ−1∑

t

(μ2t + μ′2t + 2[μtωt + μ

′t(ωt + γ)])

+(μ0 + μ′0)(−ir) + (μ1 + μ′1)(−ir − η1)

+(μ2 + μ′2)(−ir − η2) + (μ3 + μ′3)(ir − η3), (B.4)


KD ≡ 2η1rπ−1∑t

(μt + μ′t). (B.5)

Comparing to V (z) (3.26), we see that this is of the form exp(CD +KDz)V (−z). Since van Diejen requires

∑(μt+μ

′t) = 0 (cf. Eq. (3.21) in [8]),

his field w(z) is proportional to V (−z). We have written the R-factors in(B.3) in a form that makes clear how μt and μ

′t are related to our couplings

c, given by (3.21) and (3.43). Specifically, we get

μ0 = ic0, μ2 = ic1, μ′0 = ic2, μ

′2 = ic3,

μ1 = ic4, μ3 = ic5, μ′1 = ic6, μ

′3 = ic7. (B.6)

Notice that this entails (cf. also (3.45))

3∑t=0

(μt + μ′t) = 0⇔

7∑n=0

cn = 0. (B.7)

It is now straightforward (albeit very laborious) to verify that the Hamil-tonian given by Eqs. (4.1)–(4.3) in [8] is of the form C1(A + C2), whereC1, C2 are constants and A is defined by (4.7). (Observe that van Diejen’sparameter μ equals ours.) We point out that the need for omitting thefactor exp(KDz) in (B.3) in order to obtain an integrable N > 1 systemwithout the constraint

∑(μt + μ

′t) = 0 would have been hard to guess with

the bare-handed approach of [8].The integrable generalization to arbitrary couplings (obtained in [9] via

elaborate Yang-Baxter machinery) leads to arbitrary-N Hamiltonians thatare related by additive and multiplicative constant factors to the ones ofSubsection 4.2. Again, this is not at all clear by inspection. Furthermore, asit turns out, the pertinent constants are not even S8-invariant. The detailedcomparison now follows.

With a slight adaptation of notation, the BCN Hamiltonian arrived atin [9] reads

HD =N∑j=1

(Ψj(x)1/2 exp(2γ∂xj )Ψj(−x)1/2 + (x→ −x)) + Ψ0(x), (B.8)

with

Ψj(x) =

(3∏t=0

θt+1(xj − νt)θt+1(xj)

θt+1(xj + γ − νt)θt+1(xj + γ)

)

×⎛⎝∏k �=j

θ1(xj − xk + μ)θ1(xj + xk + μ)θ1(xj − xk)θ1(xj + xk)

⎞⎠ , (B.9)


Ψ0(x) =3∑t=0

(π

θ′1(0)

)2 2

θ1(μ)θ1(μ− 2γ)3∏u=0

θu+1(γ + νπtu)θu+1(νπtu)

·N∏j=1

θt+1(xj − γ + μ)θt+1(xj − γ)

θt+1(xj + γ − μ)θt+1(xj + γ)

, (B.10)

π0 = id, π1 = (01)(23), π2 = (02)(13), π3 = (03)(12). (B.11)

(We have changed β, μ, z in Eqs. (1.2)–(1.3) of [9] to γ,−μ, x, and correctedthe overall sign of Ψ0(x), cf. also Komori’s later paper [30].)

The connection of HD to H (4.6) is as follows. To start with, our pa-rameter r should be chosen equal to π, cf. (2.30). Next, the parameters νtand νt are related to our couplings c (given by (3.21) and (3.43)) via

ν0 = −ic0, ν3 = −ic1, ν0 = −ic2, ν3 = −ic3,ν1 = −ic4, ν2 = −ic5, ν1 = −ic6, ν2 = −ic7. (B.12)

With this correspondence in effect, we have

Ψj(x) = exp(πCK)Vj(−x), j = 1, . . . , N, (B.13)

Ψ0(x) = exp(πCK)[V(x) +

3∑t=0

ptEt(μ; zt)N/2R(μ− ω2)R(μ− 2γ − ω2)],

(B.14)CK ≡ −2i(N − 1)μ+ c0 + c2 + c4 + c6, (B.15)

so that

HD = exp(πCK)[H +3∑t=0

ptEt(μ; zt)N/2R(μ− ω2)R(μ− 2γ − ω2)]. (B.16)

Therefore, HD and H are related by multiplicative and additive constantsthat break most of the parameter symmetries discussed above.

We conclude this appendix with a few remarks on the verification of(B.13) and (B.14). First, (2.32) can be used to trade θ1 for s, and then (2.36)to trade s for R. Using (2.34) and (2.35), the remaining theta-functionscan also be expressed in terms of R. Then it is not hard to check (B.13).The verification of (B.14) is quite arduous, but straightforward; here, therelation (2.33) should be used as well.

Acknowledgment

We would like to thank Y. Komori for illuminating correspondence.


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ON THE PROLONGATIONOF AHIERARCHYOF

HYDRODYNAMIC CHAINS

A.B. SHABATLandau Institute for Theoretical Physics, RAS, Moscow 117334, Russia

L. MARTıNEZ ALONSODepartamento de Fısica Teorica II, Universidad ComplutenseE28040 Madrid, Spain

Abstract. The prolongation of a hierarchy of hydrodynamic chains previously studied bythe authors is presented and the properties of the differential reductions of the enlargedhierarchy are derived. Several associated nonlinear integrable models are exhibited. Inparticular, it is found that the Camassa–Holm equation can be described as a second-order differential reduction of one of the new flows included in the hierarchy.

1. Introduction

In our previous work [1], [2] we considered a hierarchy of hydrodynamicchains arising in the theory of energy-dependent Schrodinger spectral prob-lems [3], [8]. The methods we developed to analyze the properties andsolutions of this hierarchy are rather different from those used in the stan-dard Lax approach to the theory of integrable systems and revealed notonly a rich structure but also the presence of new interesting integrablenonlinear models.

An abstract formulation of this hierarchy is provided by the system ofequations[

Dn −An(λ,x)D0 , Dm −Am(λ,x)D0

]= 0, n,m ≥ 0, (1.1)

where {Dn}n≥0 denote the differentiation operators with respect to theinfinite set of independent variables x := (x0, x1, x2, . . .) and λ is an spectralparameter. Equivalently, (1.1) can be written as

DnAm −DmAn = 〈An, Am〉, n,m ≥ 1, (1.2)

263


264 A.B. SHABAT AND L. MARTıNEZ ALONSO

where

〈U, V 〉 := U(D0V )− (D0U)V.

If we assume that the functions An are n-th degree monic polynomials in λthen their coefficients are interrelated through (1.2) and by using inductionit can be proved that, up to unessential terms and without loss of generality,the functions An are determined by an infinite set of common coefficients{gm(x)}m≥1 in the form

An = λn + g1(x)λ

n−1 + · · · + gn−1(x)λ+ gn(x). (1.3)

Thus (1.1) defines the action of the derivations {Dn}n≥1 on the set ofdynamical variables {gn(x)}n≥1. This action is given by the system of flows

DnG = 〈An, G〉, n ≥ 1, (1.4)

where G = G(λ,x) is the generating function

G = 1 +g1(x)

λ+g2(x)

λ2+ · · · , λ→∞. (1.5)

In fact (1.1) is the system of compatibility conditions of (1.4). By identifyingcoefficients of powers of λ in (1.4) one gets the hierarchy of hydrodynamicchains

Dign =n∑k=1

〈gn−k, gi+k〉, n ≥ 1. (1.6)

An alternative and useful formulation of (1.4) is obtained by introducingthe generating function

H :=1

G= 1 +

h1λ+h2λ2+ · · · , λ→∞ , (1.7)

h1 = −g1, h2 = g21 − g2, h3 = −g31 + 2g1g2 − g3, . . . . (1.8)

Thus (1.4) is equivalent to

DnH = D0

(AnH

), (1.9)

which means, in particular, that the coefficients {hn(x)}n≥1 supply aninfinite set of conservation laws for (1.4).

A basic feature of (1.1) is its invariance under the replacement of thebasic set of derivation operators {Dn}n≥0 by

Dn,1 := Dn − gn(x)D0, n ≥ 1. (1.10)

A HIERARCHY OF HYDRODYNAMIC CHAINS 265

Indeed, from (1.1) one deduces (see [1], [2]) that {Dn,1}n≥1 forms a com-mutative set of operators verifying[Dn,1−An−1(λ,x)D1,1 , Dm,1−Am−1(λ,x)D1,1

]= 0, n,m ≥ 1 . (1.11)

Obviously, the process can be iterated to conclude that for all p ≥ 1[Dn,p−An−p(λ,x)Dp,p , Dm,p−Am−p(λ,x)Dp,p

]= 0, n,m ≥ p, (1.12)

whereDn,p := Dn,p−1 − gn−p+1(x)Dp−1,p−1, n ≥ p.

The goal of the present work is to enlarge the system of equations (1.1)by introducing a set of new independent variables {x−n}n≥1, its correspond-ing set {D−n}n≥1 of differentiation operators and a family {A−n(λ,x)}n≥1of polynomials in λ−1 which depend on

x := (. . . , x−1, x0, x1, . . .).

It can be proved that the functions {A−n}n≥1, as well as the functions{An}n≥1, are determined by an infinite set of common coefficients {bn(x)}n≥0in the form

A−n :=b0(x)

λn+b1(x)

λn−1+ · · ·+ bn−1(x)

λ. (1.13)

Here instead of the monic condition for {An}n≥1 we impose the vanish-ing of the constant term in the decomposition of {A−n}n≥1 in powersof λ−1. The main point is that the dependence of the dynamical vari-ables {gn(x)}n≥1

⋃{bn(x)}n≥0 on x can be described in terms of a singlegenerating function satisfying

DnG = 〈An, G〉, n ∈ Z, (1.14)

as well as

G = 1 +g1(x)

λ+g2(x)

λ2+ · · · , λ→∞,

(1.15)G = b0(x) + b1(x)λ+ b2(x)λ

2 + · · · , λ→ 0.

Nevertheless, we will treat the dynamical variables

{gn(x)}n≥1⋃{bn(x)}n≥0

as independent. Equivalently, in terms of H, the system (1.14) reads

DnH = D0

(AnH

), n ∈ Z. (1.16)


The paper is organized as follows: Section 2 is devoted to the intro-duction of several closed differential forms associated with solutions ofour enlarged hierarchy. It is also analyzed how these forms are related tointeresting symmetry transformations of the hierarchy. Section 3 presentsseveral applications arising from the consideration of the new flows includedin (1.14). In particular two 3-dimensional nonlinear models are exhibited.Furthermore, several 2-dimensional nonlinear models related to reductionsof (1.14), including the Camassa–Holm equation, are discussed.

The following notation conventions are used in this paper. Firstly, G∞and G0 stand for the expansions of G as λ→∞ and λ→ 0 , respectively.The flows associated to the variables {xn}n≥1 and {x−n}n≥1 are referred toas the positive and negative flows, respectively. We notice that the negativeflows of G0 describe the Schwarzian form of the hierarchy (1.4) (see [2]), sothat our enlarged hierarchy contains the two variants of (1.4) considered in[2].

Furthermore, we denote by V the space of formal Laurent series

V =∞∑

n=−∞anλ

n,

by Vr,s (r ≤ s) the subspaces of elements

V =s∑n=r

anλn,

and by Pr,s : V $→ Vr,s the corresponding projectors. Given V ∈ V we willalso denote (

V)r,s:= Pr,s(V ).

In particular, notice that we can write

An =(λnG∞

)0,+∞, A−n =

(λ−nG0

)−∞,−1, n ≥ 1.

2. Hodograph transformations and differential forms

The enlarged hierarchy of flows (1.14) forms a compatible system. Thatis to say, as a consequence of (1.14) the following consistency conditionsfollow

DnAm −DmAn = 〈An, Am〉, n,m ∈ Z, (2.1)

where {An}n∈Z are given by (1.3) and (1.13). For example let us prove thecompatibility between positive and negative flows. From (1.3), (1.13) and


(1.14) we deduce

D−mAn −DnA−m =(λnD−mG∞

)0,+∞ −

(λ−mDnG0

)−∞,−1

=(〈A−m, λnG∞〉

)0,+∞ −

(〈An, λ−mG0〉

)−∞,−1

=(〈A−m, An〉

)0,+∞ −

(〈An, A−m〉

)−∞,−1

= 〈A−m, An〉.The equations (2.1) represent a generalization of (1.1). Indeed they are

equivalent to the system[Dn −An(λ,x)D0 , Dm −Am(λ,x)D0

]= 0, n,m ∈ Z. (2.2)

In particular this means that we can extend the abelian set of first-orderdifferential operators (1.10) as the next theorem states.

Theorem 2.1 The family of operators

Dn,1 := Dn − gn(x)D0, n ≥ 1,D0,1 := b0D0, (2.3)

D−n,1 := D−n + bn(x)D0, n ≥ 1,forms a commutative set

[Dr,1,Ds,1] = 0, r, s ∈ Z. (2.4)

Proof. To prove (2.4) one may proceed as follows

1. For r, s ≥ 1 it is enough to set n = r,m = s and λ = 0 in (2.2)2. For r ≥ 1 and s = 0 we set to zero the coefficient of 1/λ in (2.2) withn = r and m = −1.

3. For r ≥ 1 and s ≤ −1 we equate to zero the coefficient of 1/λ in (2.2)with n = r,m = s− 1 and find

[Dr − grD0,−b−sD0] +∑I

[giD0, bjD0] = 0,

where

I = {(i, j) : i = j + r + s, 0 ≤ i ≤ r, 0 ≤ j ≤ −s− 1}.Furthermore, by setting to zero the coefficient of λ0 in (2.2) with n = rand m = s we get

[Dr − grD0,Ds] +∑I

[giD0, bjD0] = 0,


so that[Dr − grD0,Ds + b−sD0] = 0.

4. Finally, for r, s ≤ −1, equating to zero of the coefficient of 1/λ in (2.2)with n = r,m = s− 1 yields

[Dr, b−sD0] + [br−1D0,Ds−1] = 0,

and by setting to zero the coefficient of 1/λ2 in (2.2) with

n = r − 1, m = s− 1,

we deduce

[b−rD0, b−sD0]− [Dr−1, b−s−1D0] + [Ds−1, b−r−1D0] = 0.

Thus it follows at once that

[Dr + b−rD0,Ds + b−sD0] = 0.

As a consequence of this theorem we deduce

Corollary 2.2 There is a change of variables

y = (. . . , y−1, y0, y1, . . .) $→ x = (. . . , x−1, x0, x1, . . .),

determined byx0 = x0(y), xn = yn, n ∈ Z− {0}, (2.5)

where

ω1 := dx0 = b0dy0 +∑n≥1

(bndy−n − gndyn),

(2.6)

ω1 := dy0 =1

b0

(dx0 +

∑n≥1

(gndxn − bndx−n)).

Proof. Theorem 1 implies the existence of variables (yn)n∈Z such that

∂

∂yn= Dn,1, n ∈ Z. (2.7)

Hence, by taking (2.3) into account, the statement follows at once.

We notice that (2.5) is a transformation of hodograph type as it in-volves dependent and independent variables. The next result shows how thistransformation is related to the main symmetry property of our hierarchy.


Theorem 2.3 Given a solution G(λ, x) of (1.14), then it verifies

Dn,1G = 〈An−1, G〉1 n ∈ Z, (2.8)

where〈U, V 〉1 := U(D1,1V )− (D1,1U)V.

Proof. For n ≥ 0 the statement of this theorem was proved in [1], sothat we concentrate on the negative flows D−n,1, n ≥ 1. Firstly, notice that

D−n,1G = D−nG+ bn(D0G) = 〈A−n, G〉 + bn(D0G)

= λA−n−1(D0G)− (D0A−n)G. (2.9)

On the other hand

D1,1G = D1G− g1(D0G) = λ(D0G)− (D0g1)G, (2.10)

so that

D0A−n = D0

(λ−nG

)−∞,−1 =

( 1

λn+1(D1,1G+ (D0g1)G)

)−∞,−1

= D1,1A−n−1 + (D0g1)A−n−1.

Hence, by substituting in (2.9) we deduce

D−n,1G = A−n−1(D1,1G+ (D0g1)G)

−(D1,1A−n−1 + (D0g1)A−n−1)G = 〈A−n−1, G〉1.

In view of this result we may apply Theorem 1 and its corollary tothe system (2.8) and conclude the existence of a further hodograph typetransformation.

Corollary 2.4 There is a change of variables

z = (. . . , z−1, z0, z1, . . .) $→ y = (. . . , y−1, y0, y1, . . .),

determined byy1 = y1(z), yn = zn, n ∈ Z− {1}, (2.11)

where

ω2 := dy1 = b0dz1 +∑n≥1

(bndz−n+1 − gndzn+1),

ω2 := dz1 =1

b0(dy1 +

∑n≥1

(gndyn+1 − bndy−n+1)


FurthermoreDn,2G = 〈An−2, G〉2 n ∈ Z, (2.12)

where

Dn,2 =∂

∂zn, 〈U, V 〉1 := U(D2,2V )− (D2,2U)V.

Of course the process used in the above pair of theorems can be iteratedto generate an infinite chain of hodograph transformations which definesymmetries of (1.14).

According to the statement of Corollary 1 to each solution of (1.14)corresponds a closed differential form given by

ω1 := dy0 =1

b0

(dx0 +

∑n≥1

(gndxn − bndx−n)).

The next theorem provides another example of this type of correspondence

Theorem 2.5 The differential form

ω := b0dx−1 +∑n≥1

(bndx−n−1 − gndxn−1), (2.13)

is closed.

Proof. We first notice that

bn = Res( 1

λn+1G0

), gn = Res

(λn−1G∞

).

In order to prove the theorem we must verify that

∂m−1gn = ∂n−1gm, n,m ≥ 1, (2.14)

∂−m−1bn = ∂−n−1bm, n,m ≥ 0, (2.15)

∂m−1bn = −∂−n−1gm, n ≥ 0,m ≥ 1. (2.16)

Let us start with (2.14), we have

∂m−1gn − ∂n−1gm =

Res(〈(λm−1G∞)0,∞ , λn−1G∞〉

−〈(λn−1G∞)0,∞ , λm−1G∞〉)=

Res(〈λm−1G∞ , (λn−1G∞)−∞,−1〉

−〈(λn−1G∞)0,∞ , λm−1G∞〉)=

Res(〈λm−1G∞, λn−1G∞〉

)= 0


Similarly, (2.15) can be proved as follows

∂−m−1bn − ∂−n−1bm =

Res(〈(λ−m−1G0)−∞,−1 , λ−n−1G0〉 =

−〈(λ−n−1G0)−∞,−1 , λ−m−1G0〉)=

Res(〈λ−m−1G0 , (λ

−n−1G0)0,∞〉

−〈(λ−n−1G0)−∞,−1 , λ−m−1G0〉)=

Res(〈λ−m−1G0, λ

−n−1G0〉)= 0.

Finally, (2.16) is proved in the form

∂m−1bn + ∂−n−1gm =

Res(〈(λm−1G∞)0,∞ , λ−n−1G0〉

+〈(λ−n−1G0)−∞,−1 , λm−1G∞〉)=

Res(〈(λm−1G∞)0,∞ , (λ−n−1G0)−∞,−1〉

+〈(λ−n−1G0)−∞,−1 , (λm−1G∞)0,∞〉)= 0.

3. Applications

3.1. 3-DIMENSIONAL INTEGRABLE MODELS

The differential forms introduced in the above section reveal the presenceof interesting 3-dimensional integrable models in the hierarchy (1.14) as thenext examples show.

Example 1The x1-flow of (1.14) implies at once that

D1g1 = D0g2.

Moreover, from (2.5) and (2.6) we have that the potential function

y0 = Q(x),

satisfies

g1 =D1Q

D0Q, g2 =

D2Q

D0Q,


and therefore, if we denote x := x0, y := x2 and z := x1 we get(QzQx

)z=

(QyQx

)x. (3.1)

Example 2By identifying the coefficient of λ0 in the equation for the first negative

flow

D−1G = 〈b0λ,G〉,

we getD−1b0 = 〈b0, b1〉.

On the other hand , from Theorem 3 if we introduce the function

Q = Q(x)

which satisfies

dQ = ω = b0dx−1 +∑n≥1

(bndx−n−1 − gndxn−1),

thenb0 = D−1Q, b1 = D−2Q.

Hence, by denoting x := x0, y := x−2 and z := x−1, we obtain

Qzz = QzQxy −QxzQy. (3.2)

3.2. 2-DIMENSIONAL INTEGRABLE MODELS

In [1],[2] we developed a theory of reductions of the hierarchy (1.4) basedon imposing differential constraints on G ≈ G∞ of the form(

F(λ,G,Gx, Gxx, . . .))−∞,−1 = 0, x := x0. (3.3)

In particular the following three classes of reductions associated to arbitrarypolynomials a = a(λ) in λ were characterized:

Zero-order reductions

They are defined by constraints of the form

a(λ)G = U(λ,x), U :=(a(λ)G

)0,+∞, (3.4)


First-order reductions

They are characterized by the differential constraints

Gx + a(λ) = U(λ,x)G, U :=( aG

)0,+∞, (3.5)

or, equivalently,

−Hx + a(λ)H2 = U(λ,x)H, U :=(aH

)0,+∞. (3.6)

Second-order reductions

They are determined by constraints of the form

1

2GGxx −

1

4G2x + a(λ) = U(λ,x)G

2, U :=( aG2

)0,+∞. (3.7)

Equivalently, in terms of the function H, we may rewrite (3.7) as

{Dx,H}+ a(λ)H2 = U(λ,x), U :=(aH2

)0,+∞. (3.8)

Here we are denoting

{Dx,H} :=3

4

(HxH

)2

− 1

2

HxxH.

Under the differential constraints (3.8) the hierarchy (1.14) describesthe KdV hierarchy and its generalizations associated to energy-dependentSchrodinger spectral problems. Indeed, if we define the functions ψ(λ,x)by

ψ(λ,x) := exp(D−1x φ), φ := −12

HxH±√a(λ)H, (3.9)

then from (1.16) and (3.8) it is straightforward to deduce that

Dnψ = −12(Dn logH)ψ ±

√aAnHψ

= An(− 1

2Dx logH ±

√aH

)ψ − 1

2An,xψ

= Anψx −1

2An,xψ,

ψxx = (φx + φ2)ψ = ({Dx,H}+ aH2)ψ = Uψ.

In other words, the functions ψ are wave functions for the integrablehierarchies associated to energy-dependent Schrodinger problems. The


evolution law of the potential function U under the flows (1.14) can bedetermined from the equation

DnU = −12An,xxx + 2UAn,x + UxAn, (3.10)

which arise as an straightforward consequence of (1.14) and (3.7).

Theorem 3.1 The differential constraints (3.4) (for a given a(λ) such thata(0) �= 0), (3.5) and (3.7) are left invariant by the enlarged hierarchy (1.14).

Proof. Let us indicate the proof by considering the second-order case

F :=1

2

GxxG− 1

4

G2x

G2+a(λ)

G2, x := x0, (3.11)

where a(λ) is a N -degree polynomial in λ. After the analysis of [1], [2], itremains to prove that this constraint is invariant under the negative flowsof (1.14). To this end, let us consider

DtG = 〈A−n, G〉, t := x−n, n ≥ 1.

Then, one finds

DtF =∂F∂GDtG+

∂F∂Gx

DxDtG+∂F∂Gxx

D2xDtG

= −12D3xA−n + 2(DxA−n)F +A−nDxF .

Now, the constraint associated with (3.11) implies

F = U = λN + λN−1uN−1(x) + · · · u0(x), (3.12)

where

U(λ,x) :=

(a(λ)

G2

)0,+∞

.

In this way we have

DtF = −12D3xA−n + 2UDxA−n + UxA−n. (3.13)

Notice that despite of the fact that these differential constraints areformulated for G ≈ G∞, because of the polynomial character of the right-hand side of (3.12) we may analytically extend (3.12) to become a constraintfor both G ≈ G∞ or G ≈ G0. Thus, (3.12) implies

1

2G0G0,xx −

1

2G2

0,x + a(λ) = UG20,


and by differentiating with respect to x one obtains

−12G0,xxx + 2UG0,x + UxG0 = 0,

which by taking into account that

A−n =(λ−nG0

)−∞,−1,

yields

DtF = −12D3xA−n + 2UDxA−n + UxA−n

=1

2

(λ−nD3

xG0

)0,+∞ − 2U

(λ−MDxG0

)0,+∞ − Ux

(λ−nG0

)0,+∞.

It proves that (DtF

)−∞,−1 = 0.

It is important to analyze the action of the transformation (2.5) on thereductions of our hierarchy. To this end we observe that from (2.10) wehave (x := x0)

Gx =1

λ(D1,1G+ g1,xG),

Gxx =1

λ2

(D2

1,1G+ g1,xD1,1G+ (λg1,xx + g2,xx − g1g1,xx)G),

and as a consequence one readily finds that the elements of the classes offirst-order and second-order reductions transform under (2.5) as

−Hx + a(λ)H2 = UH $→ −D1,1H + a(λ)H2 = UH,(3.14)

U :=(aH

)0,+∞ = λU + h1,x,

wherea(λ) := λa(λ),

and

{Dx,H}+ a(λ)H2 = U $→ {D1,1,H}+ a(λ)H2 = U ,(3.15)

U :=(aH2

)0,+∞ = λ2U +

1

2λh1,xx +

1

2h2,xx −

3

4(h1,x)

2 − 1

2h1h1,xx,

wherea(λ) := λ2a(λ),


respectively. This means that there are three basic cases, namely: those cor-responding to linear a(λ) for first-order reductions, and linear and quadratica(λ) for second-order reductions. They are associated with the Burgers,KdV (Korteweg–deVries) and NLS (Nonlinear Schrodinger) hierarchies,respectively.

The Burgers hierarchy

If we impose the differential constraint

−D0H + λH2 = UH, U = λ+ h1, (3.16)

then the positive flows of the hierarchy (2.1) reduce to the Burgers hierarchy[2]. For example, by inserting the expansion of H as λ→∞ in the x1 flow

D1H = D0

((λ− h1)H

), (3.17)

it follows

D1h1 = D0h2 −D0(h21).

Moreover (3.16) yields h2 = D0h1, so that we get Burgers equation

ut = uxx − (ux)2, t := x1, x := x0, u := h1.

Notice that by inserting the expansion of H at λ = 0 in (3.17) and (3.16)we get

h1 =axa, a :=

1

b0,

and

at = axx,

which constitutes the Hopf-Cole transformation between the Heat equationand Burgers equation.

According to (3.14) the change of variables (2.5) transforms (3.16) into

−D1,1H + λ2H = (λ2 + λh1 + h2)H, (3.18)

and the first positive flow of the transformed hierarchy becomes

D2,1H = D1,1

((λ− h1)H

). (3.19)

Thus, by substituting the expansions of H as λ → 0 and λ → ∞ in theseequations we get

D1,1a = −h2a, D2,1a = −D1,1(h1a), D2,1h1 = D1,1(h2 − h21).


Hence, the following nonlinear model results{ut = −(uv)x,vt = −(log u)xx − 2vvx

(3.20)

Herex := y1, t := y2, u := a, v := h1.

It is interesting to rewrite the system (3.20) in terms of the potentialfunctions associated to the differential forms (2.7) and (2.12). Indeed, from(2.7) if we take x0(y) := −Q(y) it follows that

h1 = −Qx, (log a)x = −h2 = Qt − (Qx)2.

Hence the second equation of the system (3.20) becomes trivial and thefirst equation adopts the form

Qtt = Qxxx + 3QxQxt + (Qt − 3Q2x)Qxx. (3.21)

This nonlinear model is the first member of a family of integrable systemswith unusual dispersion law [9]. Alternatively, from (2.12) if we introducethe function z1 = Q(y) we have

a = Qx, h1 = −QtQx,

so that the first equation of (3.20) becomes trivial, while the second equa-tion writes (

QtQx

)t= (logQx)xx +

((QtQx

)2 )x. (3.22)

The KdV hierarchy

Let us consider the differential constraint (3.8) with

a(λ) = λ+ ε.

It can be formulated as

{Dx,H}+ (λ+ ε)H2 = U, U := λ+ ε+ 2h1. (3.23)

By using (3.10) it follows that the first positive flow is given by the KdVequation

ut =1

4uxxx −

3

2uux +

1

2ε ux, u := 2h1. (3.24)

Now, if we use the variable y := y0 of (2.5), (2.6)

Dy = D01 = b0D0,


and denote

H := b0H, (3.25)

then (3.23) implies at once that

{Dy, H}+ (λ+ ε)H2 = λb20 + ε. (3.26)

On the other hand, from the x−1-flow

D−1G = 〈b0λ,G〉,

we deduce

D−1H =(b0λH)x, D−1b = 〈b0, b1〉.

Thus, it readily follows that

DtH = Dy

((b0λ+ b1

)H

), (3.27)

where

t := y−1, Dt = D−1,1 = D−1 + b1D0.

By identifying the coefficient of λ in (3.26) as λ→ 0 we get

b20 = −1

2uyy + 2εu+ 1, u := −b1

b0. (3.28)

Moreover, identifying the coefficient of λ0 in (3.27) as λ→∞ yields

b0,t = b1,y. (3.29)

Hence, if we differentiate (3.28) with respect to t and use (3.28), (3.29) weget

2b0b0,t = 2b0b1,y = −2b0b0,yu− 2b20uy,and

2ε ut −1

2uyyt + 6ε uuy −

1

2uuyyy − uyyuy + 2uy = 0, (3.30)

which is the Camassa–Holm equation [10]

The NLS hierarchy

The quadratic case of the second-order differential reduction (3.8)

{Dx,H}+ (λ2 + ε)H2 = U(λ,x), U := λ2 + 2h1λ+2h2 + h21 + ε, (3.31)


corresponds to the NLS hierarchy [4]. This fact can be seen by noticingthat the Zakharov–Shabat spectral problem

Ψx =

(λ uv −λ

)Ψ, Ψ :=

(ψ1ψ2

), (3.32)

leads to a pair of Schrodinger spectral problems with quadratic dependenceon λ

φxx = (λ2 − λDx(log u) + {Dx, u} + uv)φ, ψ1 =√uφ,

(3.33)ψxx = (λ2 + λDx(log v) + {Dx, v}+ uv)ψ, ψ2 =

√vψ.

Thus there are two different ways of relating (3.31) and (3.32), which aregiven by

2h1 = −Dx log u, 2h2 − h1,x + ε = uv, (3.34)

and2h1 = Dx log v, 2h2 + h1,x + ε = uv. (3.35)

However, in both cases the first positive flow of the hierarchy (1.14) con-strained by (3.31)

D1H = D0

((λ− h1)H

),

yields the coupled NLS system (t := x1, x := x0){ut =

12uxx − u2v + ε u,

vt = −12vxx + v

2u− ε v(3.36)

Now let us proceed as in the KdV case by using again the variabley := y0 of (2.5), (2.6)

Dy = D01 = b0D0,

and denoting H := b0H. Then (3.31) implies

{Dy, H}+ (λ2 + ε)H2 = U , U := λ2b20 + 2λb2h1 + ε. (3.37)

Notice that by identifying the coefficient of λ in (3.37) as λ→ 0 we get

2b20h1 = −1

2uyy + 2εu, u := −b1

b0. (3.38)

On the other hand we have

DtH = Dy

((b0λ+ b1

)H

), (3.39)

wheret := y−1, Dt = D−1,1 = D−1 + b1D0,


which leads to the following equation for U

Ut = −1

2Ayyy + 2UAy + UyA, A :=

1

λ− u. (3.40)

Then, by identifying the coefficients of λ and λ2 in (3.41) we obtain thefollowing analogue of the Camassa–Holm equation for the quadratic caseof the second-order constraints{

vt = −(vu)y,wt = −wyyy + (v2)y − uwy + 2wuy,

(3.41)

Here we are denoting

v := b0, w := −1

2uyy + 2εu.

Acknowledgements

A.B. Shabat was supported by the Russian Foundation for Basic Research(Grant Nos. 00-15-96007-L and 01-01-00874-A). L. Martınez Alonso wassupported by the DGCYT project BFM2002-01607.

References

1. L. Martınez Alonso and A. B. Shabat, Phys.Lett. A 300, 58 (2002).2. L. Martınez Alonso and A. B. Shabat, J. Non. Math. Phys. 10, 229 (2003).3. V. G. Mikhalev, Funct. Anal. Appl. 26 No 2, 140 (1992).4. M. Jaulent and I. Miodek, Lett. Math. Phys. 1, 243 (1976); Lett. Nuovo Cimento

20, 655 (1977).5. L. Martınez Alonso, J. Math. Phys. 21, 2342 (1980).6. M. Antonowicz and A. P. Fordy, Physica 28 D, 345 (1987).7. A. Shabat, Universal models of solitonic hierarchies. To appear in Theor. Math.

Phys., June 2003.8. A. N. W. Hone, Phys. Lett. A 249, 46 (1998).9. R. Heredero, V. Sokolov and A. B. Shabat, A new class of linearizable equations, to

be published.10. R. Camassa and D. Holm, Phys. Rev. Lett. 71, 1661 (1993).

SUPERINTEGRABLE SYSTEMS IN CLASSICAL AND

QUANTUM MECHANICS

P. WINTERNITZ ([email protected])Centre de recherches mathematiques etDepartement de mathematiques et statistique,Universite de Montreal, C.P. 6128, succ. Centre-Ville,Montreal, QC, H3C 3J7, Canada

Abstract. A brief review is given of the status of superintegrability, i.e., the theoryof classical and quantum mechanical finite-dimensional systems with more integrals ofmotion than degrees of freedom. Typically, in classical mechanics such systems are char-acterized by periodic motion, in quantum mechanics their energy levels can be calculatedalgebraically.

1. Introduction

Conceptually speaking, an integrable system is a system for which one canmake global, or long term, predictions. To express this in more technicalterms, let us first consider a Hamiltonian system in classical mechanics,described by a Hamiltonian function H(�x, �p), depending on n coordinates�x = (x1, . . . , xn) and n canonically conjugated momenta �p = (p1, . . . , pn).Such a system is called integrable, or Liouville integrable, if there exist nwell defined functions on phase space Xi(�x, �p) (including the HamiltonianH) satisfying the following properties:

dXidt

= {Xi,H} = 0 (1.1)

{Xi,Xj} = 0 (1.2)

rank∂(X1, . . . ,Xn)

∂(x1, . . . , xn, p1, . . . , pn)= n, (1.3)

where { , } denotes the Poisson bracket. In other words, a Hamiltoniansystem with n degrees of freedom is integrable if there exist n functionallyindependent well defined integrals of motion in involution [1, 2].

281


282 P. WINTERNITZ

A Hamiltonian system is superintegrable if it is integrable and thereexist k further integrals Ya(�x, �p) (with 1 ≤ k ≤ n− 1) satisfying

dYadt

= {Ya,H} = 0, a = 1, . . . , k; (1.4)

rank∂(X1, . . . ,Xn, Y1, . . . , Yk)

∂(x1, . . . , xn, p1, . . . , pn)= n+ k. (1.5)

In general we have {Xi, Ya} �= 0, i.e., only n of the n + k integrals arein involution. If the total number of independent integrals of motion isN = 2n − 1, the system is called maximally superintegrable. If we haveN = n+ 1, it is minimally superintegrable.

Integrability and superintegrability have physical consequences. Indeed,if a system is integrable, the equations

X1 = H = E, Xi = ci, i = 2, . . . , n (1.6)

confine the motion to an n-dimensional torus in phase space (possibly anoncompact version of a torus). If a system is maximally superintegrablethe additional equations Ya = ka (a = 1, . . . , n − 1) further constrainthe motion to a line in phase space (E, ka, ci are constants), in particularthe trajectories are completely determined. It follows that for a classicalmaximally superintegrable system all finite trajectories are periodic.

The best known maximally superintegrable Hamiltonian systems are theKepler–Coulomb system and the isotropic harmonic oscillator, described bythe potentials

VK(r) =α

r, VH(r) = ω

2r2, (1.7)

respectively. Indeed, in addition to the angular momentum �L, the Kepler–Coulomb problem admits the Laplace–Runge–Lenz vector

�A = �L× �p+ α

r3�r (1.8)

and the harmonic oscillator admits the tensor

Tik =1

2pipk + ω

2xixk, (1.9)

as integrals of motion.A classical theorem due to Bertrand [3] states that the potentials (1.7)

are the only two rotationally invariant ones for which all finite trajectoriesare closed.

In quantum mechanics the situation is quite analogous. A system de-scribed by the Schrodinger equation

Hψ = Eψ, H = −12�2Δ+ V (�x) (1.10)

SUPERINTEGRABLE SYSTEMS 283

is superintegrable if there exists a set of Hermitian operators {X1, . . . ,Xn,Y1, . . . , Yk} (with H = X1 and 1 ≤ k ≤ n− 1) satisfying

[H,Xi] = 0, [H,Ya] = 0 , (1.11)

[Xi,Xj ] = 0 ; (1.12)

i = 1, . . . , n, a = 1, . . . k .

The operators {Xi, Ya} must satisfy an independence condition (thatneeds a rigorous definition with the correct classical limit). The physicalconsequences of superintegrability in quantum mechanics are interesting.In particular, since in general we have [Xi, Ya] �= 0, the integrals of motiongenerate a non-Abelian Lie algebra (finite or infinite-dimensional). Thisin turn implies that the energy levels of such systems will be degenerate.Indeed, the accidental degeneracy of the hydrogen atom and the harmonicoscillator, described respectively by their O(4) and SU(3) symmetries, areexamples of this phenomenon [4–6].

Another consequence of superintegrability is at this stage conjectured,rather than proven. The conjecture is that all maximally superintegrablequantum systems are exactly solvable, i.e., their energy levels can be calcu-lated algebraically and their bound state wave functions, in an appropriategauge and in appropriate coordinates, are polynomials. This conjecture isborne out by many examples [7–9].

We see that superintegrable systems in both classical and quantummechanics have interesting mathematical and physical properties. Suchsystems are rare and it is worthwhile to search for them systematically.

To formulate the problem of finding superintegrable systems mathemat-ically, some restrictions must be imposed and some choices made.

First of all one must choose the space one is working in: a Euclideanspace En, a space of constant nonzero curvature, or a more general Rie-mannian, pseudo Riemannian, or complex Riemannian space.

Secondly, the form of the Hamiltonian must be specified, involving ascalar potential, or a scalar and vector one. Alternatively, one could considera many-particle Hamiltonian, conceivably involving particles with spin.

Finally restrictions must be imposed on the form of the integrals ofmotion. Usually they are postulated to be polynomials of a definite orderin the momenta, often just second-order polynomials.

The purpose of this report is to provide a brief review of the status ofsuperintegrability in classical and quantum mechanics. In Section 2 we con-centrate on quadratic integrability, i.e., the case of a standard Hamiltonianand integrals of motion, all of second order in the momenta. The case ofthird-order integrals is discussed in Section 3. We limit ourselves to twodimensional spaces though the extension to n dimensions is immediate, atleast conceptually.

284 P. WINTERNITZ

2. Quadratic superintegrability

2.1. EUCLIDEAN SPACES

The systematic search for superintegrable systems started quite some timeago [10, 11]. The approach was via commuting operators in quantum me-chanics in two-dimensional Euclidean space with the Hamiltonian

H = −12Δ + V (x, y) (2.1)

and the operators commuting with H were assumed to be of second order

X = f ik(�x)pipk + gi(�x)pi + φ(�x). (2.2)

The form of the Hamiltonian (2.1) is left invariant by the group E(2) ofEuclidean transformations, which do however change the potential V (x, y).

The corresponding Lie algebra e(2) has a basis given by

L3 = y∂x − x∂y, P1 = ∂x, P2 = ∂y. (2.3)

The commutation relation [H,Xa] = 0 implies very strong restrictions onthe functions f ika and gia, namely we find:

X = aL23 + b(L3P1 + P1L3) + c(L3P2 + P2L3) +

d(P 21 − P 2

2 ) + 2eP1P2 + αL3 + βP1 + γP2 + φ(x, y), (2.4)

where a, . . . , e and α, β, γ are constants. Furthermore, the potential V (x, y),the function φ(x, y) and the constants in (2.4) must satisfy:

[α(y∂x − x∂y) + β∂x + γ∂y]V = 0, (2.5)

φx = −2(ay2 + 2by + d)Vx + 2(axy + bx− cy − e)Vy,

φy = 2(axy + bx− cy − e)Vx + 2(−ax2 + 2cx+ d)Vy.(2.6)

The compatibility condition φxy = φyx provides a further equation for thepotential, namely

(−axy − bx+ cy + e)(Vxx − Vyy) + [a(x2 − y2)− 2by − 2cx− 2d]Vxy − 3(ay + b)Vx + 3(ax− c)Vy = 0. (2.7)

Eq. (2.5) is the condition for a first-order operator commuting with theHamiltonian to exist. This is the same as requiring that V (x, y) be invariantunder rotations, or translations. For α �= 0 this is equivalent to puttingV = V (r). For α = 0, β2 + γ2 �= 0 this is equivalent to putting V = V (x).To find potentials V (x, y) allowing only a second-order operator X we put


α = β = γ = 0. We can simplify the operator X (and Eq. (2.7)) byEuclidean transformations and linear combinations with the HamiltonianH:

X → X = λgXg−1 + μH, λ �= 0, g ∈ E(2). (2.8)

The transformation (2.8) admits two invariants

I1 = a, I2 = (2ad − b2 + c2)2 + 4(ae− bc)2. (2.9)

If we have I1 = I2 = 0, there is a third invariant, namely I3 = d2 + e2.Depending on the values of these invariants we can transform X and thepotential V (x, y) into precisely one of the following forms:

1. I1 = I2 = 0, I3 �= 0

XC = −1

2(P 2

1 − P 22 ) + f(x)− g(y), V = f(x) + g(y). (2.10)

2. I1 �= 0, I2 = 0

XR = L23 − 2g(θ), V = f(r) +

1

r2g(θ), (2.11)

x = r cos θ, y = r sin θ.

3. I1 = 0, I2 �= 0

XP = L3P2 + P2L3 +g(η)ξ2 − f(ξ)η2

ξ2 + η2, V =

f(ξ) + g(η)

ξ2 + η2, (2.12)

x =1

2(ξ2 − η2), y = ξη.

4. I1 �= 0, I2 �= 0

XE = L23 +

l2

2(P 2

1 − P 22 )− l2

cosh 2ρf(σ) + cos 2σg(ρ)

cos2 σ − cosh2 ρ, l > 0

V =f(σ) + g(ρ)

cos2 σ − cosh2 ρ, (2.13)

x = l cosh ρ cos σ, y = l sinh ρ sinσ.

We see that a second-order operator X commuting with the Hamiltonian Hin E2 exists if and only if the Schrodinger equation allows the separation ofvariables in cartesian, polar, parabolic, or elliptic coordinates. These corre-spond to the cases (2.10), (2.11), (2.12) and (2.13), respectively (l in (2.13)is the focal distance and we have l2 = I2/I

21 . Thus all quadratically integrable

systems (2.1) are also separable. They allow (multiplicative) separationof variables in the Schrodinger equation and also (additive) separation ofvariables in the Hamilton–Jacobi equation.

286 P. WINTERNITZ

The close relation between quadratic integrability and separation ofvariables persists in other spaces (En for arbitrary n) and also in spaces ofconstant curvature, as well as much more general Riemannian and pseudo-Riemannian spaces.

The system will be quadratically superintegrable if it allows two integralsof the form (2.2). It will then also be multiseparable, i.e., allow the sepa-ration of variables in at least two coordinate systems. Four such systemswere shown to exist in E2. The corresponding potentials are:

VI = ω2(x2 + y2) +α

x2+β

y2, (2.14)

VII = ω2(4x2 + y2) +α

y2+ βx, (2.15)

VIII =α

r+

1

2r2

(β1

1 + cos θ+

β21− cos θ

)=

1

ξ2 + η2

(2α +

β1ξ2+β2η2

), (2.16)

VIV =α

r+

1√2r

(β cos

θ

2+ γ sin

θ

2

)=

1

ξ2 + η2(2α + βξ + γη). (2.17)

Classical trajectories, wave functions and energy spectra for these fourpotentials were determined long ago [10, 11]. More recently, it was shownthat these superintegrable systems are invariant under generalized Lie sym-metries, and that they allow recursion operators [12].

Let us now demonstrate the exact solvability of these four superinte-grable systems [7]. First of all we note that the generalized hydrogen atompotentials VIII and VIV can be reduced to the case of the generalized har-monic oscillator VI. Indeed, the Schrodinger equation for VIII in paraboliccoordinates is{

−12

1

ξ2 + η2(∂2ξ + ∂

2η) +

1

ξ2 + η2

(2α+

β1ξ2+β2η2

)}ψ = Eψ. (2.18)

This can be rewritten as{−12(∂2ξ + ∂

2η)− E(ξ2 + η2) +

β1ξ2+β2η2

}ψ = −2αψ. (2.19)

Similarly, the Schrodinger equation for VIV can be rewritten as{−12(∂2ξ + ∂

2η)− E

[(ξ − β

2E

)2

+

(η − γ

2E

)2]}ψ =(

−2α− β2 + γ2

4E

)ψ. (2.20)


Changing notation, we write the Schrodinger equation for the potential VIas

HI

(x, y;

ω2

2,A

2,B

2

)ψ = (2.21){

−12(∂2x + ∂

2y) +

ω2

2(x2 + y2) +

1

2

(A

x2+B

y2

)}ψ = Eψ.

Equations (2.19) and (2.20) can be rewritten as

QIIIψ = −2αψ, QIII = HI(ξ, η,−E, β1, β2) (2.22)

and

QIVψ =

(−2α− β

2 + γ2

4E

)ψ, QIV = HI

(ξ − β

2E, η − γ

2E,−E, 0, 0

).

(2.23)This means that the Schrodinger equation for potentials VIII and VIV hasbeen reduced to that for potential VI, with a metamorphosis of the couplingconstant [13]: the energy and coupling constant ω are interchanged. Thusit suffices to show that the generalized harmonic oscillator (2.21) is exactlysolvable. To see this, we solve eq. (2.21) and obtain:

ψnm(x, y) = xp1yp2e−

ω2(x2+y2)L

(− 12+p1)

n (ωx2)L(− 1

2+p2)

m (ωy2), (2.24)

A = p1(p1 − 1), B = p2(p2 − 1),

where L(α)k (z) is a Laguerre polynomial. The ground state wave function

ψ00(x, y) = xp1yp2e−

ω2(x2+y2) (2.25)

figures in eq. (2.24) as a universal factor for all bound states. We use itto perform a gauge rotation of the Hamiltonian HI and simultaneouslytransform to new variables t = ωx2, u = ωy2. We obtain

hI =1

ωψ−100 HIψ00

∣∣∣∣t=ωx2,u=ωy2

= −2t∂2t − 2u∂2u + 2t∂t + 2u∂u −

(2p1 + 1)∂t − (2p2 + 1)∂u + 1 + p1 + p2. (2.26)

Thus the transformed Hamiltonian hI is an element of the enveloping alge-bra of the affine Lie algebra aff(2, IR), realized by the vector fields

J1 = ∂t, J2 = ∂u, J3 = t∂t, J4 = u∂u, J5 = u∂t, J6 = t∂u.(2.27)

288 P. WINTERNITZ

For hI the Schrodinger equation reduces to

hIPN (t, u) = (n+m)PN (t, u), N = n+m, (2.28)

where PN (t, u) are polynomials of order n +m. The operators commutingwith HI are

XC = −12(P 2

1 − P 22 ) +

ω2

2(x2 − y2) + 1

2

(A

x2− By2

),

XR = L23 −

A

cos2 θ− B

sin2 θ. (2.29)

After a gauge rotation with the gauge factor ψ00(x, y) and a change ofvariables to t and u we can write hI, xc = ψ

−100 XCψ00 and xR = ψ

−100 XRψ00

as:

hI = −2J3J1 − 2J4J2 + 2J3 + 2J4 − (2p1 + 1)J1 − (2p2 + 1)J2,xC = 2J3J1 − 2J4J2 − 2J3 + 2J4 + (2p1 + 1)J1 − (2p2 + 1)J2,xR = 4J3J5 + 2J4J6 − 8J3J4 + 2(2p1 + 1)J5 − 2(2p2 + 1)J3 −

2(2p1 + 1)J4 + 2(2p2 + 1)J6. (2.30)

All three integrals of motion hI, xC and xR are in the enveloping alge-bra of aff(2, IR). This algebra contains no raising operators: only loweringand reproducing ones. Hence all three operator (2.30) preserve the flag ofpolynomials

PN (t, u) = {tmun : 0 ≤ m+ n ≤ N}. (2.31)

This is the origin of the exact solvability of the superintegrable HamiltonianHI. The arguments for the generalized anisotropic harmonic oscillator HII

are very similar [7]. The generalized hydrogen atoms HIII and HIV arereduced to HI by the coupling constant metamorphosis (2.22), or (2.23),respectively.

A systematic study of quadratically superintegrable systems in three-dimensional Euclidean space E3 was also started in the 1960s [14] and con-tinued in the 1990s [15, 16]. Superintegrable and exactly solvable systemsalso exist in En for any n [9, 17].

2.2. SPACES OF NONZERO CONSTANT CURVATURE

Quadratic superintegrable systems have been classified and analyzed onreal spheres Sn and real Lorentzian hyperboloidsHN , as well as on complexEuclidean spaces and spheres. For lack of space we shall not reproduce theresults here, but merely refer to the original articles.

The two dimensional cases S2, H2, E2(C| ) and S2(C| ) have been treatedin detail [18–20]. Higher-dimensional spaces, more sporadically [21–24].


2.3. TWO DIMENSIONAL SPACES OF NONCONSTANT CURVATURE:DARBOUX SPACES

Let us consider a two-dimensional space with infinitesimal distance

ds2 =2∑

i,j=1

gij(u)duiduj. (2.32)

The classical Hamiltonian in this space is

H =1

2gij(u)pipj + V (u), (2.33)

and the Schrodinger equation is written as:

Hψ =

[−12

1√g∂ui√ggik∂uk + V (u)

]ψ = Eψ, g = det gik. (2.34)

In keeping with our general approach, we are looking for a metric gij(u) andpotentials V (u) such that the Hamiltonian H admits first, or second-orderintegrals of motion

X(1) = ai(u)pi, X(2) = aij(u)pipj + φ(u), (2.35)

dX(i)

dt= {X(i),H} = 0. (2.36)

First of all we consider the case V (u) = 0, φ(u) = 0. Eq. (2.36) thendetermines Killing vectors ai(u), or Killing tensors aij(u). The problem offinding all complex two-dimensional Riemannian spaces with more thanone quadratic integral of motion was completely solved by Darboux [25]and Koenigs [26]. They established the following results

1. A two-dimensional Riemannian space can admit 0, 1, or 3 Killing vec-tors. If it admits one it is a space of revolution. If it admits three, it isa space of constant curvature.

2. A two-dimensional Riemannian space can admit 0, 1, 2, 3, or 5 Killingtensors. If it admits 5, then it is a space of constant curvature and allthe Killing tensors are reducible: they are bilinear expressions in the 3Killing vectors. If the space admits 3 Killing tensors, then one of themis reducible, i.e., it is the square of a Killing vector, and the space is aspace of revolution.

3. They gave a complete classification of spaces with 2 and 3 Killingtensors.

290 P. WINTERNITZ

4. Four types of Riemannian spaces with 3 Killing tensors exist. Thecorresponding infinitesimal distances are given by:

I. ds2 = (x+ y)dxdy,

II. ds2 =

(a

(x− y)2 + b)dxdy,

III. ds2 =(ae−(x+y)/2 + be−(x+y)

)dxdy,

IV. ds2 =a(e(x−y)/2 + e−(x−y)/2) + b[e(x−y)/2 − e−(x−y)/2]2 dxdy.

(2.37)

We shall call these spaces Darboux spaces of type I,. . . ,IV. Superinte-grable systems exist in all four of them [8, 27]. Here we shall discuss thefirst space only and consider a real form of it. We put x = u+iv, y = u−iv.The infinitesimal distance and Hamiltonian reduces to

ds2 = 2u(du2 + dv2), H =1

4u(p2u + p

2v), (2.38)

respectively. The three classical integrals of motion (for the case of zeropotential) are

K = pv, X1 = pupv −v

2u(p2u + p

2v), (2.39)

X2 = pv(vpu − upv)−v2

4u(p2u + p

2v).

The Killing vector K generates a one-dimensional group of isometries ofthe space. The most general second-order integral of motion will have theform

X = αX1 + βX2 + γK2. (2.40)

Acting on X with the transformation g = exp(αK) we find that X canbe transformed into precisely one of the following three representatives oforbits:

L1 = X1 + aK2, L2 = X2 + aK

2, L3 = K2. (2.41)

Let us now find coordinates that will allow the separation of variablesin the Hamilton—Jacobi equation

H ≡ 1

2gik∂S

∂ui∂S

∂uk= E. (2.42)

To do this we add a further equation, given by the integral X:

X ≡ aik ∂S∂ui

∂S

∂uk= λ, (2.43)


where X is one of the expressions (2.41). The separable coordinates are theeigenvalues ρ1, ρ2 of the characteristic equation

|aik − ρgik| = 0, (2.44)

under the condition that they are distinct, i.e., ρ1 �= ρ2. Once separablecoordinates are found, we can add a potential V (ρ1, ρ2) that does notspoil the separation. In general, the classical quadratically integrable andseparable system will have the form

H =1

σ(α) + τ(β)[p2α + p

2β + f(α) + g(β)], (2.45)

L =1

σ(α) + τ(β)[σ(α)(p2β + g(β)) − τ(β)(p2α + f(α))]. (2.46)

The system will be superintegrable if it allows a further integral, indepen-dent of the pair {H,L}. By necessity it will have the form (2.40).

For all details we refer to the original article [8]. Here we just statethe result, namely: there exist precisely three quadratically superintegrablesystems in the Darboux space of type I. They are all separable in at leasttwo coordinates systems. Their Hamiltonians and integrals of motion are

1) H1 =1

4u[p2u + p

2v + a1 + a2v + a3(u

2 + v2)],

R1 = X1 +1

2u[−a1v + a2(u2 − v2) + a2v(u2 − v2)], (2.47)

R2 = K2 + a2v + a3v2.

2) H2 =1

4u

[p2u + p

2v + b1(4u

2 + v2) + b2 +b3v2

],

R1 = X2 −1

4u

[b1v

4 + b2 + b34u2 + v2

v2

], (2.48)

R2 = K2 + b1v2 +

b3v2.

3) H3 =1

u(p2u + p

2v + a),

R1 = X1 −av

2u,

R2 = X2 −av2

4u, (2.49)

R3 = K.

The quantum integrable systems are obtained by substituting

pu → −i�∂

∂u, pv → −i�

∂

∂v

292 P. WINTERNITZ

in all of the above expressions.All of the corresponding quantum systems are exactly solvable. Indeed,

if we multiply the Schrodinger equation by the denominator u and inter-change the energy E with a1 for H1, with b2 for H2 and with a for H3,we obtain a shifted isotropic harmonic oscillator, a generalized anisotropicharmonic oscillator and a linear potential, respectively. Thus the abovesimple metamorphosis of the coupling constants takes each of the abovesystems into a superintegrable system in E2 that has been shown to beexactly solvable [7].

3. Superintegrability with a third-order integral of motion

Quadratic superintegrability is by now rather well understood, at least forscalar potentials. The main features that have emerged are the following

1. Integrable and superintegrable systems with integrals of motion that arelinear, or quadratic in the momenta coincide in classical and quantummechanics.

2. Integrable systems in n dimensions with n quadratic (or linear) com-muting integrals of motion are separable: the Hamilton-Jacobi andSchrodinger equations allow the separation of variables. Superintegrablesystems are multiseparable.

3. All known maximally superintegrable systems are exactly solvable

Much less is known about integrable systems with higher-order integralsof motion. One systematic study is due to Drach, [28] who consideredclassical integrable systems in two-dimensional complex Euclidean spaceE(2, C| ). He found 10 such systems with third-order integrals (in classicalmechanics). It was shown by Ranada [29] (see also Tsiganov [30]) that 7 ofthese systems are actually reducible. They are quadratically superintegrableand the third-order integral is the commutator of two second-order ones.

A classification of all integrable classical and quantum systems withthird-order integrals is a difficult task, even in two-dimensional Euclideanspace. Indeed, let us start from the quantum case, setting

H = −12�2Δ+ V (x, y), (3.1)

X =∑

0≤j+k≤3[fjk(x, y)p

j1pk2 + p

j1pk2fjk(x, y)], pk = −i�

∂

∂xk.

The operator X contains terms of order 3, 2, 1 and 0 in the momenta, andhas been symmetrized to ensure that it is Hermitian. The commutativitycondition [H,X] = 0 implies that even and odd terms in the momentamust commute separately with H. Thus, if we require integrability with a


third-order integral, rather than superintegrability, we must put fjk = 0 forj + k = 2 and also f00 = 0. Furthermore, the third-order terms must lie inthe enveloping algebra of e(2). This is true for any potential V (x, y). Thepotential will figure in four remaining determining equations. Explicitly, weobtain [31]

X =∑

i+j+k=3

Aijk(Li3pj1pk2 + p

j1pk2Li3) + g1(x, y)p1 + p1g1(x, y)

+ g2(x, y)p2 + p2g2(x, y), (3.2)

where Aijk are constants. The remaining determining equations are

g1,x = 3f1Vx + f2Vy, (3.3)

g2,y = f3Vx + 3f4Vy, (3.4)

g1,y + g2,x = 2(f2Vx + f3Vy), (3.5)

g1Vx + g2Vy −�2

4[f1Vxxx + f2Vxxy + f3Vxyy + f4Vyyy

+ 8A300(xVy − yVx) + 2(A210Vx +A201Vy)] = 0. (3.6)

The functions fi are given in terms of the constants Aijk as

f1 = −A300y3 +A210y

2 −A120y +A030,

f2 = 3A300xy2 − 2A210xy +A201y

2 +A120x−A111y +A021,

f3 = −3A300x2y +A210x

2 −A201xy +A111x−A102y +A012,

f4 = A300x3 +A201x

2 +A102x+A003.

(3.7)

Thus we have an overdetermined system of 4 partial differential equationsfor 3 functions V (x, y), g1(x, y) and g2(x, y).

The first thing to notice about the system (3.3–(3.6) is that the Planckconstant � figures in eq. (3.6). Hence the classical and quantum case differ,contrary to the case of quadratic integrability. Indeed, in the classical caseeqs. (3.3), (3.4) and (3.5) are the same, but eq. (3.6) reduces to

g1Vx + g2Vy = 0. (3.8)

The difference between classical and quantum integrability was alreadynoticed by Hietarinta [32, 33].

Another difference between the cases of second- and third-order integralsof motion is that the system (3.3)–(3.6) (or (3.7)) is overdetermined, andhence gives rise to compatibility conditions. Eqs. (3.3), (3.4) and (3.5) implya linear equation for the potential, namely

f3Vxxx + (3f4 − 2f2)Vxxy + (3f1 − 2f2)Vxyy + f2Vyyy

294 P. WINTERNITZ

+ 2(f3,x − f2,y)Vxx + 2(3f1,y + 3f4,x − f2,x − f3,y)Vxy+ 2(f2,y − f3,x)Vyy + (3f1,yy + 3f3,xx − 2f2,xy)Vx+ (f2,yy + 3f4,xx − 2f3,xy)Vy = 0. (3.9)

Further compatibility conditions for the potential are nonlinear. This iswhat makes the problem difficult to solve, even in the classical case. Thenonlinear compatibility conditions are third-order equations in the classicalcase, fifth-order ones in quantum mechanics.

Let us now look at a simpler problem, namely that of superintegrabilitywith one first-order integral of motion and one third-order one. We considerthe classical and quantum cases separately. In both cases, the first-orderintegral can be either X = P2 or X = L3. The corresponding potentialsatisfies V = V (x), or V = V (r) respectively.

In the classical case, no new superintegrable systems emerge. For X =P2, we obtain two potentials

V1 = ax, V2 =a

x2. (3.10)

Both allow second-order integrals, in addition to P2. The third-order inte-grals are products of P2 and the second-order ones.

For X = L3 we once again get the Kepler–Coulomb potential and theharmonic oscillator, as in eq. (1.7). The third-order integrals are againreducible.

In the quantum case, the terms in eq. (3.6) proportional to �2 makeall the difference. Indeed, taking X1 = P2 and a third-order integral of theform

X2 = L3P21 + P

21L3 + g1P1 + P1g1 + g2P2 + P2 g2, (3.11)

we find that these two integrals are allowed by any potential V (x) satisfying

�2V 2x = 4(V − V1)(V − V2)(V − V3), (3.12)

where Vi are constants. If they are all different, we obtain a finite or asingular solution in terms of Jacobi elliptic functions:

V (x) = �2ω2k2 sn2(ωx, k), V (x) =

�2ω2

2(1 + cn(ωx, k)),

or V (x) =�2ω2

sn2(ωx, k). (3.13)

In the case of multiple roots, V1 < V2 = V3, or V1 = V2 < V3 we obtainsolutions in terms of elementary functions:

V (x) =�2ω2k2

cosh2 ωx, V (x) =

�2ω2

sin2 ωx. (3.14)


For V1 = V2 = V3 we reobtain a known potential, V (x) = �2/x2, however,

with additional integrals of motion. In general, the third-order integral(3.11) is equal to

X1 = L3P21 + P

21L3 + {−y(3V − α), P1}+

{2xV − αx+∫V dx, P2}, (3.15)

where { · , · } denotes the anticommutator. Commuting X1 with X2 weobtain a further integral

X3 = P31 + {3V − α,P1}. (3.16)

For X1 = L3, i.e., rotationally invariant potentials, the classical andquantum cases coincide. In fact, no new superintegrable potentials emerge.The only ones are VK = α/r2 and V = ω2r2, and the third-order quantumintegrals are commutators of lower-order ones.

4. Conclusions

Finite-dimensional superintegrable systems are of considerable interest intheir own right, because of their mathematical and physical properties.Moreover, an intriguing relationship with soliton theory, i.e., the theoryof infinite-dimensional integrable systems, is emerging. First of all, a su-perintegrable system can be characterized by the fact that it allows anon-Abelian algebra of integrals of motion with at least one n-dimensionalAbelian subalgebra. The situation is similar for soliton equations, e.g.,the Korteweg-de Vries and the Kadomtsev-Petviashvili equations. Theirintegrals of motion form infinite-dimensional non-Abelian algebras. Thesehave Abelian subalgebras that are also infinite-dimensional. In this sense,soliton equations are superintegrable, rather than just integrable.

A further relation between soliton theory and superintegrable systemsappears in the study of quantum superintegrability in E2 space with third-order integrals of motion. In Section 3 we saw that elliptic function (3.13)and soliton (3.14) solutions of the KdV equation figure as superintegrablepotentials. The relation between solutions of the KdV and superintegrabil-ity in E2 goes further. When superintegrable systems allowing one third-order and one second-order integral are considered, Painleve transcendentsemerge as the corresponding potentials [34].

Acknowledgements

This text was written during the author’s visit to the Departamento deFısica Teorica II of the Universidad Complutense of Madrid. The author

296 P. WINTERNITZ

thanks the Department for its hospitality, and specially Miguel A. Rodrıguezfor helpful discussions. The author’s research is partially supported by re-search grants from NSERC of Canada, FQRNT du Quebec and the NATOcollaborative grant PST.CLG 978431.

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