Point Interactions in Nonrelativistic Quantum Mechanics

Bachelor's thesis

Point Interactions in

Nonrelativistic Quantum

Mechanics

Simone Carlo Surace

Institute for Theoretical Physics, Bern

Supervisor: Prof. Uwe-Jens Wiese

January 5, 2010

iii

Abstract

In this work, point interactions in non-relativistic (Schrödinger)quantum mechanics are discussed in one to three dimensions. Inaddition to the treatment using the theory of self-adjoint exten-sions, two regularizations are presented. The results with respectto the spectral properties from both approaches are compared.

Contents

1 Introduction 2

1.1 Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Self-adjoint extensions of point interaction Hamiltonians 4

2.1 Theory of self-adjoint extensions . . . . . . . . . . . . . . . . . . . . 42.2 One-dimensionional point interaction . . . . . . . . . . . . . . . . . . 5

2.2.1 Four-parameter family of self-adjoint extensions . . . . . . . . 52.2.2 Spectral properties . . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Parity eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Two- and three-dimensional point interactions . . . . . . . . . . . . . 102.3.1 One-parameter families of self-adjoint extensions . . . . . . . 102.3.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Regularizations of contact interactions 14

3.1 Point-splitting regularization . . . . . . . . . . . . . . . . . . . . . . 143.1.1 δ interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.2 δ′ interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.3 δ′′ interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Dimensional regularization . . . . . . . . . . . . . . . . . . . . . . . . 213.2.1 Some preliminary calculations . . . . . . . . . . . . . . . . . . 213.2.2 δ interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.3 ∆δ interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Bibliography 29

1

Chapter 1

Introduction

Point or contact interactions stand for a class of models which describe the motion ofa particle experiencing no force except at one or several points in the con�gurationspace which represent inpurities of a very small size. The most direct and exactlysolvable implementation of such an interaction can be achieved by means of theDirac delta function. By removing the point where the interaction occurs fromthe con�guration space, one arrives at a well de�ned symmetric operator whose self-adjoint extensions can be interpreted as contact interactions with di�erent properties.

In addition to the powerful and complete treatment provided by the theory ofself-adjoint extensions which we present in Chapter 2, there are several procedureswhich are supposed to yield equivalent results. One is the approximation by localshort-range interactions which is not discussed here but in[1] and similar works. Thesecond type encompasses deliberately heuristic procedures such as the point-splittingand dimensional regularization, which are discussed in Chapter 3. They are inspiredby physical intuition in an attempt to apply the concept of renormalization to extractmeaningful physical results from an initially divergent theory. To some extent, thisapproach works, but one is presented with several counterintuitive results. Therelation between the results of the two treatments are discussed in the remarksfound throughout Chapter 3.

1.1 Historical remarks

In this section, we brie�y review the development of some ideas which relate tocontact interactions and provide some references which are suggested for furtherreading.

The history of contact interactions dates back to the early days of quantummechanics. Kronig and Penney used a crystal lattice of rectangular potentials within�nitesimal width in their historical paper [10]. Later, short-range potentials wereused to model the interaction between nucleons, namely in the deuteron and tritonmodels. Thomas [13] established the concept of renormalization of the coupling, andFermi [8] used the models in the description of nucleons in hydrogenated substances,thereby introducing velocity-dependent potentials, called Fermi pseudopotentials.All these methods lead to a characterization via some boundary condition in the locus

2

Introduction 3

of the interaction, but it was not until 1961 that this fact was put on a mathematicalbasis by Berezin and Faddeev [4]. They were the �rst to study singular rank oneperturbations to the free Schrödinger operator, a topic which is covered extensivelyin the monograph [2].

�eba[14] discovered that in one dimension, there is a much larger family of contactinteractions than in higher dimensions. The family of generalized point interactionsin one dimension has four parameters and permits up to two bound states.

The context of Schrödinger contact interactions was greatly enlarged by the ob-servation that they are equivalent to the non-relativistic limit of a suitable quantum�eld theory, e.g. a λφ4 model. This topic is discussed in[6] and the references therein.

Finally, for an extensive review and a long list of references to the subject, wesuggest [1] and the recent monograph [2].

Chapter 2

Self-adjoint extensions of point

interaction Hamiltonians

In this chapter, we want to present the results on contact interactions which areobtained by a careful de�nition of the Hamiltonian as a self-adjoint operator. Forthat end, we �rst take a look at some of the basic axioms of quantum mechanics.

In quantum mechanics, the state of a system is described by a time-dependentvector |Ψ(t)〉 in a separable Hilbert space H [11]. The dynamics of the system aregiven in terms of the Schrödinger equation

id

dt|Ψ(t)〉 = H|Ψ(t)〉 (2.1)

where H is the Hamiltonian which represents the total energy of the system andwhich is the generator of the one-parameter unitary group U(t) = e−iHt of timeevolution. In order to accomplish this task, the Hamiltonian is required to be aself-adjoint operator1 de�ned on a dense subspace D(H) ⊂ H, called the domain ofH.

2.1 Theory of self-adjoint extensions

Usually, the form of the Hamiltonian is derived from some correspondence principleor similar considerations. It is necessary to �nd a domain on which this expressionconstitutes a well-de�ned and symmetric operator H0. In general, however, thisoperator is not self-adjoint. Instead, its adjoint H†0 may turn out to have a largerdomain than H0. We may de�ne the de�ciency subspaces

K±.= ker(i∓H†0) ⊂ H

A famous theorem due to von Neumann (see [12], p.140) gives a characterizationof symmetric operators in terms of the numbers n± = dim(K± ), called de�ciencyindices.

1For an operator T to be self-adjoint, it must be symmetric, i.e. 〈φ|Tψ〉 = 〈Tφ|ψ〉 for allφ, ψ ∈ D(T ) and its domain must coincide with the domain of its adjoint T †.

4

Self-adjoint extensions of point interaction Hamiltonians 5

1. If n+ = n− = 0, the de�ciency subspaces are trivial. H0 is then said to beessentially self-adjoint, meaning that there is a unique self-adjoint extension ofH0, given by the closure of H0.2.

2. If at least one of the two de�ciency indices is not zero, H0 is not essentiallyself-adjoint, anda) There are self-adjoint extensions of H0 if and only if the two de�ciency in-

dices are equal. The self-adjoint extensions are parametrized by the unitarymaps from K+ onto K−.

b) If one of the de�ciency indices is zero and the other one is not, there areno non-trivial symmetric extensions of H0. There is no hope to give theoperator rigorous meaning within the framework of quantum mechanics.

Through the extension to a self-adjoint operator, a symmetric operator acquiresquantum completeness. This implies that the time evolution of any initial state isuniquely speci�ed. In general, if the initial operator is not essentially self-adjoint,each self-adjoint extension leads to di�erent dynamics. Therefore, every symmetricoperator which is supposed to represent a quantum observable is subject to theaforementioned analysis.

2.2 One-dimensionional point interaction

In one dimension we de�ne the one-center point interaction by considering3 theoperator

H0 = − 12m

d2

dx2(2.2)

with domain D(H0) ={ψ ∈W 2,2(R)|ψ(0) = ∂ψ(0) = 0

}⊂ L2(R). It can be shown

by integration by parts that H0 is symmetric. However, H†0 is de�ned on the largerdomain W 2,2(R), where R = R\{0} denotes the pointed real line. It is obvious thatthe domain of H0 is too small, while the restrictions on the functions in D(H†0) aretoo loose. The goal is now to �nd some domain where the restrictions are distributedequally between H and H†.

2.2.1 Four-parameter family of self-adjoint extensions

In order to determine the self-adjoint extensions of H0, we have to �nd the de�ciencysubspaces, hence all the solutions to the equations

H†0φ± = − 12m

d2

dx2φ± = ± iφ± (2.3)

2The closure of any densely de�ned operator T is obtained by closing its graph with respect tothe topology of H×H.

3This derivation closely follows [1, 14].


which are given, up to normalization, by linear combinations of the two orthogonalfunctions4

φ1± (x) =

{eix√± 2mi, x > 0

0, x < 0, φ2

± (x) =

{0, x > 0e−ix

√± 2mi, x < 0

(2.4)

where√

2mi .= ω and√−2mi .= iω are chosen to have positive imaginary part.

Therefore, the de�ciency indices are (2, 2). For each unitary map U ∈ U(2) there isa self-adjoint extension

D(HU ) = {ψ0 + ψ+ + Uψ+|ψ0 ∈ D(H0) ∧ ψ+ ∈ K+}HU (ψ0 + ψ+ + Uψ+) = H0ψ0 + iψ+ − iUψ+

(2.5)

By the decomposition U(2) ' SU(2)×U(1) ' S3×S1, this is a four-parameterfamily of self-adjoint extensions which may be parametrized by

U = eiϕ

(u1 u2

−u∗2 u∗1

), ϕ ∈ [0, π), u1u

∗1 + u2u

∗2 = 1 (2.6)

A function ψ ∈ D(HU ) thus takes the form

ψ = ψ0 + αφ1+ + βφ2

+ + eiϕ(u1α+ u2β)φ1− + eiϕ(−u∗2α+ u∗1β)φ2

− (2.7)

where α, β ∈ C. Using equation (2.4) and the de�nition of D(H0), we can write

limx↑0

[ψ(x)

∂ψ(x)

]=

(−eiϕu∗2 1 + eiϕu∗1

−ωeiϕu∗2 ω(eiϕu∗1 − i)

)(α

β

).= T1

(α

β

)(2.8)

limx↓0

[ψ(x)

∂ψ(x)

]=

(1 + eiϕu1 eiϕu2

−ω(eiϕu1 − i) −ωeiϕu2

)(α

β

).= T2

(α

β

)(2.9)

or5

limx↓0

[ψ(x)

∂ψ(x)

]= T2 ·T−1

1 · limx↑0

[ψ(x)

∂ψ(x)

](2.10)

4Note that functions from D(H†0) may be discontinuous at the origin because the origin does notbelong to their support.

5If U is diagonal, or equivalently, if u2 = 0, then T2 and T1 are both singular. This is a specialdecoupled case for which the boundary conditions are not expressible in the form (2.10). Al-though we do not further discuss it, ref. [7] o�ers several parametrizations which include thiscase.


Thus, with a few exceptions each self-adjoint extension can be characterized by a setof two boundary conditions on the functions in its domain. Since

det(T2 ·T−11 ) =

detT2

detT1= −u2

u∗2

.= e2iθ, θ ∈ [0, π) (2.11)

we can write T2 ·T−11 = eiθM with detM = 1. Moreover, from the relation

e−iθM∗ =(T2 ·T−1

1

)∗ = e−iθM (2.12)

we can deduce that M is a real matrix. Thus, the boundary conditions can bewritten as

limx↓0

[ψ(x)

∂ψ(x)

]= eiθ

(a b

c d

)limx↑0

[ψ(x)

∂ψ(x)

](2.13)

where a, b, c, d ∈ R, θ ∈ [0, π) and ad − bc = 1. By Hθ,M we denote the operatorwhich is de�ned on the subspace of functions from W 2,2(R) satisfying (2.13).

2.2.2 Spectral properties

Since Hθ,M is now self-adjoint, its spectrum is a subset of the real numbers. Let us�rst determine the point spectrum. For κ > 0 the eigenvalue equation

Hθ,Mψ = − 12m

d2ψ

dx2= − κ2

2mψ, ψ ∈ D(Hθ,M ) (2.14)

has the solution

ψ(x) =

{Ae−κx, x > 0Beκx, x < 0

(2.15)

where A and B are determined by the boundary conditions which characterize Hθ,M .Hence,

A

(1

−κ

)= BeiθM

(1

κ

)⇔

(1 −eiθ(a+ bκ)

κ eiθ(c+ dκ)

)(A

B

)= 0 (2.16)

and the condition for nontrivial solutions A, B is a quadratic equation

bκ2 + (a+ d)κ+ c = 0 (2.17)

Thus, there may be one or two bound state energies depending on the parametersof the matrix M . We do not list all the particular cases at this point. Instead, we


proceed to the continuous spectrum, where we only consider plane waves of the form

Ψl(x) =

{eikx +Rle

−ikx, x < 0Tle

ikx, x > 0Ψr(x) =

{Tre−ikx, x < 0

e−ikx +Rreikx, x > 0

(2.18)

for k > 0. The coe�cients are again determined by the boundary conditions(Tl

ikTl

)= eiθM

(1 +Rl

ik − ikRl

)⇔

(1 −eiθ(a− ikb)ik −eiθ(c− ikd)

)(Tl

Rl

)= eiθ

(a+ ikb

c+ ikd

)(2.19)(

−1−Rrik − ikRr

)= eiθM

(−TrikTr

)⇔

(−eiθ(a− ikb) 1

−eiθ(c− ikd) ik

)(Tr

Rr

)=

(−1

ik

)(2.20)

They read

Tl =2ikeiθ

bk2 + (a+ d)ik − c, Rl =

bk2 − (a− d)ik + c

bk2 + (a+ d)ik − c(2.21)

Tr =2ike−iθ

bk2 + (a+ d)ik − c, Rr =

bk2 + (a− d)ik + c

bk2 + (a+ d)ik − c(2.22)

and satisfy |Tl|2 + |Rl|2 = |Tr|2 + |Rr|2 = 1.

2.2.3 Parity eigenstates

Now we would like to determine which subclass of self-adjoint extensions permitsbound states which are also parity eigenstates, i.e. for which

(Pψ)(x) = ψ(−x) = χψ(x), χ = ± 1 (2.23)

Naturally for such functions, we have (Pψ)′(x) = −χψ′(x), and thus

χ limx↑0

[ψ(x)

−∂ψ(x)

]!= limx↓0

[ψ(x)

∂ψ(x)

]= eiθ

(a b

c d

)limx↑0

[ψ(x)

∂ψ(x)

](2.24)

This translates to two systems of equations for the wave function and its derivativeon the negative half-line(

aeiθ − χ beiθ

ceiθ deiθ + χ

)limx↑0

[ψ(x)

∂ψ(x)

]= 0, χ = ± 1 (2.25)


κ case 1 case 2 case 3a b

case 4

a = 0b ≷ 0

b = 0, a = ± 1ac < 0

b 6= 0, |a| ≤ 1, a ≷ 0ab > 0 ∨ ab < 0

b 6= 0, |a| > 1, a ≷ 0ab < 0

0

χ = ± 1|c|

˛c2

˛ χ = ± 1

1−|a||b|

χ = ± 1

χ = ∓ 11+|a||b|

χ0 = ± 1 |a|−1|b|

χ1 = ∓ 1 |a|+1|b|

Figure 2.1: The four cases of bound parity eigenstates arising in the family of self-adjoint

extensions. The absolute value of the depicted energy levels and the relative

position between di�erent cases is irrelevant.

The parameters of the self-adjoint extensions have to be chosen such that the deter-minants of the above matrices vanish, i.e. the two relations

(a− d)eiθ = 1− e2iθ

(d− a)eiθ = 1− e2iθ(2.26)

must hold. They imply

a = d, e2iθ = 1⇔ θ = 0 and χ = a+ bκ (2.27)

and obviously from equation (2.16) we obtain A = χB for the coe�cients of thebound state wave functions. For this choice of parameters, from equation (2.17) thepossible bound state energies (see �g. 2.1) reduce to

1. If a = 0, then bc = −1 and there is one bound state with κ = |c| and parityχ = sign(b).

2. If b = 0 we deduce from ad−bc = 1 that a = ± 1. If ac < 0, there is one boundstate with κ =

∣∣ c2

∣∣ and χ = a. Otherwise, there is no bound state.3. If b 6= 0 and |a| ≤ 1, there is only one bound state.

a) If ab > 0 and |a| 6= 1, then κ = 1−|a||b| and χ = sign(a) = sign(b).

b) If ab < 0, then κ = 1+|a||b| and χ = −sign(a) = sign(b).

4. If b 6= 0 and |a| > 1, there are two bound states if ab < 0. The ground state hasthe energy κ0 = |a|−1

|b| and parity χ0 = sign(a) = −sign(b) and the �rst excited

state lies at κ1 = |a|+1|b| and has parity χ1 = −sign(a) = sign(b). If ab > 0, there

is no bound state.


The transmission and re�ection coe�cients become parity invariant as well and read

Tl = Tr =2ik

bk2 + 2aik − c, Rl = Rr =

bk2 + c

bk2 + 2aik − c(2.28)

2.2.4 Remarks

The contact interaction in one dimension has very rich properties. We �rst note thatthe well-known δ interaction, de�ned by the formal Hamiltonian H = − 1

2md2

dx2 +V0δ(x), corresponds to the parameters a = d = 1, b = θ = 0 and c = 2mV0 (seeSection 3.1.1) and has a single bound state of even parity if V0 < 0.

There are various choices of parameters which produce interactions with singlebound states of even or odd parity and various scattering properties, e.g. for bc < 0,which is possible in the cases 1,3 and 4, the potential becomes completely transparentat k = |c/b|1/2. Moreover, it is interesting that there are parameters for which thereare two bound states. In particular, the ground state may be of odd parity, whichis usually impossible in quantum mechanics due to the assumption that the wavefunction has to be continuous. However, the boundary conditions arising with con-tact interactions permit discontinuities of both the wave function and its derivativeat x = 0, thus leading to a new class of physical models.

2.3 Two- and three-dimensional point interactions

In the Hilbert space HD = L2(RD), D = 2, 3, we start with the operator

D(H0) = C∞0 (RD), H0ψ = − 12m

∆Dψ (2.29)

where ∆D = ∂i∂i is the Laplacian in D dimensions and RD = RD\{0}. We denote

by H0 its closure in L2(RD). The adjoint may be written as

D(H†0) ={φ ∈W 2,2

loc (RD) ∩ L2(RD)|∆Dφ ∈ L2(RD)}

H†0φ = − 12m

∆Dφ(2.30)

(see [1] and the references therein). Thus, H0 represents a symmetric but non-self-adjoint operator.

2.3.1 One-parameter families of self-adjoint extensions

Instead of deriving the self-adjoint extensions of H0 in a similar way as in onedimension by solving the equations

H†0φ± = − 12m

∆Dφ± = ± iφ± (2.31)


directly, we �rst introduce polar or spherical coordinates and use the separation ofvariables to write

D = 2 : ψ(~x) = R(r)eimθ, m ∈ Z (2.32)

D = 3 : ψ(~x) = R(r)Y ml (θ, ϕ), l = 0, 1, 2, . . . ; m = −l,−l + 1, . . . , l (2.33)

where Y ml denote the spherical harmonics. The Laplacians in two and three dimen-

sions read

∆2 =1r

∂

∂r

(r∂

∂r

)+

1r2

∂2

∂θ2(2.34)

∆3 =1r2

∂

∂r

(r2∂

∂r

)+

1r2

[1

sin θ∂

∂θ

(sin θ

∂

∂θ

)+

1sin2 θ

∂2

∂ϕ2

](2.35)

Here we can use the identities

1r

d

dr

(rdR(r)dr

)=

1r1/2

d2

dr2

(r1/2R(r)

)+R(r)4r2

(2.36)

1r2

d

dr

(r2dR(r)dr

)=

1r

d2

dr2(rR(r)) (2.37)

to express the Hamiltonian in terms of the reduced 'radial' operator components

D = 2 : hm2 = − d2

dr2+m2 − 4−1

r2, m ∈ Z (2.38)

D = 3 : hl3 = − d2

dr2+l(l + 1)r2

, l = 0, 1, 2, . . . (2.39)

de�ned on suitable domains (see [1] for details). The operators hm2 and hl3, l,m 6= 0turn out to be self-adjoint in L2((0,∞)) and as such, they describe a free particlewith non-zero angular momentum. We know from physical considerations that sucha system permits no bound states. In other words, we can state that the partialwave decompositions of the free kinetic energy operator − 1

2m∆ and the operator H0

agree in the channels where l,m 6= 0.

For l,m = 0, however, they are not the same. If they were self-adjoint, h02 and

h03 would describe free particles with zero angular momentum. However, this is not

the case, since they each have de�ciency indices (1, 1) and therefore a one-parameterfamily of self-adjoint extensions. We obtain a characterization of the self-adjoint


extensions of H0 in terms of boundary conditions which are imposed only on s-waves

D = 2 : D(Hα) ={ψ ∈ D(H†0) | α lim

r↓0

ψ(r)ln r

= limr↓0

[limr′↓0

ψ(r′)ln r′

ln r − ψ(r)]}

Hαψ = − 12m

∆2ψ, −∞ < α ≤ ∞(2.40)

D = 3 : D(Hα) ={ψ ∈ D(H†0) | lim

r↓0

[ψ(r) + rψ′(r)

]= α lim

r↓0rψ(r)

}Hαψ = − 1

2m∆3ψ, −∞ < α ≤ ∞

(2.41)

By the same argument, we expect that there are no bound states of non-zero angularmomentum. In fact, the normalized functions

D = 2 : ψ(~x) =κ√πK0(κ|~x|) (2.42)

D = 3 : ψ(~x) =κ1/2

(4π2)1/4e−κ|~x|

|~x|(2.43)

for κ > 0 (where K0 denotes the modi�ed Bessel function of the second kind) arethe only solutions of

Hαψ = − κ2

2mψ (2.44)

Applying the conditions from equation (2.41) to the functions (2.43), we obtain

D = 2 : κ = 2e−γ−α, α ∈ R (2.45)

D = 3 : κ = −α, α < 0 (2.46)

where γ ≈ 0.5772 is Euler's constant. We can see that in two dimensions, there isa bound state except when α =∞, which corresponds to the free Hamiltonian H∞.In three dimensions, there is a bound state only if α < 0.

2.3.2 Remarks

In contrast to the one-dimensional case, the contact interactions in two and threedimensions are much simpler, each having only a one-parameter family of self-adjointextensions. Each extension demands a slightly di�erent behaviour of the wave func-tions for small |~x|. The extension parameter is proportional to the inverse scatteringlength (c.f [1]) and also determines the value of the binding energy.

We brie�y mention that in D ≥ 4, the Laplacian from equation (2.29) is essen-tially self-adjoint, its closure being the free Hamiltonian.


Obviously, the set of possible contact interactions is tied to the topology of thecon�guration space. In one dimension, removing the origin makes of the initiallyconnected con�guration space R an unconnected space R. In two dimensions, thespace remains connected but its �rst fundamental group π1 changes, i.e. closedcurves which contain the origin are no longer contractible (π1 isomorphic to Z). Fora general Riemannian manifold M and the associated Laplace-Beltrami operator∆M , the relations between the topology of M and the self-adjoint extensions of ∆M

are discussed in [3]. The results presented therein can be applied also to relativisticsettings.

Chapter 3

Regularizations of contact interactions

In this chapter, we present the treatment of the Dirac delta potential in one dimen-sion using standard methods. On this basis, two regularizations of contact interac-tions are discussed. The �rst is an attempt at producing a singular interaction withinternal structure. This is done by using linear combinations of δ potentials, whichis called 'Point-splitting Regularization'.

The second regularization extends the concept of dimension to the whole complexplane in order to 'smooth out' some of the divergent integrals commonly encounteredin momentum space. Applying this so called 'Dimensional Regularization', whichis also known from perturbative quantum �eld theory, to the problem of quantummechanics, we are able to derive (at least partially) the contact interactions in D =2, 3 and a subclass of the generalized point interaction in one dimension.

Both regularizations require renormalization of the coupling strength. In abroader context, renormalization is a procedure which is paramount in quantum�eld theories. The fact that it can be applied (and in fact is required) in the contextof quantum mechanics illustrates the close relationship between the two frameworks.This has been made rigorous e.g. for free particles and in two dimensions (see [6]).

3.1 Point-splitting regularization

In the point-splitting regularization we use the fact that the problem of a singleDirac delta potential in one dimension can be solved explicitly (see Section 3.1.1).Expressions of the form δ(n) are then replaced by appropriate linear combinationsof δ corresponding to discretized forms of derivatives, regulated by a parameter rwhich relates to the spacing of the δ functions.

Additionally, a variable coupling (as a function of r) is introduced in order toallow for renormalization. Without renormalization, the binding energy of the systemis in�nite, thus a zero renormalization of the coupling absorbs this in�nity. The limitr → 0 is taken afterwards.

In the following, the well-known solution of the single Dirac delta potential ispresented. On this basis the �rst and second derivatives are discussed.

14

Regularizations of contact interactions 15

3.1.1 δ interaction

The Schrödinger equation

− 12m

d2ψ(x)dx2

+ V0δ(x)ψ(x) = Eψ(x) (3.1)

can be formally integrated1 over (−ε, ε)

0 =− 12m

ε∫−ε

d2ψ(x)dx2

dx+ V0

ε∫−ε

δ(x)ψ(x)dx− Eε∫−ε

ψ(x)dx (3.2)

In the limit ε→ 0 one obtains a boundary condition at the origin

∂ψ+(0)− ∂ψ−(0) = 2mV0ψ(0) (3.3)

The derivative of the wave function is discontinuous at the origin while it is assumedthat the wave function itself is continuous. Since apart from the singular interactionat the origin, the particle moves freely, the only physically reasonable ansatz for thebound state is

ψ(x) = Ae−κ|x| =

{Aeκx, x < 0Ae−κx, x > 0

(3.4)

where κ is assumed to be positive and A is a normalization factor. This yieldsE = −EB = − κ2

2m for the binding energy. The boundary condition in equation (3.3)reads

−2κ = 2mV0 (3.5)

Therefore, the coupling constant V0 must be negative in order to admit a boundstate. Since the derivative of the wave function includes a step function, it can beexpressed by an inde�nite integral

ψ′(x) =−Aκe−κ|x|sign(x) = Aκe−κ|x|[1− 2θ(x)]

=Aκe−κ|x|(

1− 2∫ x

−∞δ(x′)dx′

), x 6= 0

(3.6)

This shows that the second derivative is a δ function balancing the singular termfrom the potential in equation (3.1).

1This calculation is formal since ψ cannot be restricted to belong to the space of test functions S.As it turns out, ψ is not even continuously di�erentiable.


One can further look for scattering solutions. Because of parity symmetry of thepotential it is su�cient to consider plane waves of the form

Ψ(x) =

{eikx +Re−ikx, x < 0Teikx, x > 0

(3.7)

where k > 0 is the wave number and E(k) = k2

2m > 0 is the energy of the wave.Again, we assume that the wave function is continuous at the origin. Along with theboundary condition in equation (3.3), this yields an inhomogeneous system of linearequations

T −R = 1 (3.8)(1− 2mV0

ik

)T +R = 1 (3.9)

The transmission and re�ection coe�cients read, respectively

T =ik

ik + κ=

12

(eiφ + 1

), R =

−κik + κ

=12

(eiφ − 1

), (3.10)

where

φ = sign(κ) arccos(k2 − κ2

k2 + κ2

)and κ = −mV0. This notation is convenient because it shows that T and R arelocated in the complex plane on circles of radius 1

2 centered at ± 12 respectively.

At low energies, for k → 0, this results in φ = ±π, T = 0 and R = −1,which means that the δ potential is totally re�ective, leading to a standing wavewhich vanishes at x = 0. At high energies φ ≡ 0 mod 2π, R = 0 and T = 1,corresponding to a perfectly transparent potential.

3.1.2 δ′ interaction

By replacing the δ′ potential by the lowest order approximation2 of the derivative,the Schrödinger equation may be rewritten as

− 12m

d2ψ(x)dx2

+ Vrδ(x+ r)− δ(x− r)

2rψ(x) = Eψ(x) (3.11)

2Higher order approximations are obtained by taking Taylor expansions at an increasing numberof sampling points and solving for the �rst derivative.


where Vr is a variable coupling constant which will be necessary in order to get a�nite binding energy. For the bound state the following ansatz is used

ψ(x) =

Aeκx, x < −rBe−κx, x > r

Ceκx +De−κx, −r < x < r

(3.12)

By virtue of the procedure developed in Section 3.1.1 in the case of a single δ poten-tial, the two boundary conditions

∂ψ+(± r)− ∂ψ−(± r) = ∓ 2mVr2rψ(± r) (3.13)

along with the two continuity conditions, form a linear system of equations0

(mVrr − κ

)e−rκ −κeκr κe−κr

−(mVrr + κ

)e−κr 0 κe−κr −κerκ

e−κr 0 −e−κr −eκr

0 e−κr −eκr −e−κr

A

B

C

D

= 0 (3.14)

to which nontrivial solutions exist only if the determinant vanishes

κ2 + (e−4κr − 1)m2V 2

r

4r2= 0

⇐⇒ 4κr = − log[1− 4κ2r2

m2V 2r

]=

4κ2r2

m2V 2r

+O(r4)(3.15)

In anticipation of the limit r → 0 only the term of leading order is retained. Thecoupling constant has to be renormalized such that the quantity

κ =m2V 2

r

r(3.16)

remains �nite as r tends to zero. The appropriate function turns out to be Vr ∼ r1/2,whereas any function rx, x > 1

2 would imply κ = 0. Since we are interested in boundstates, we assume that the renormalized coupling is of the form

Vr = V0

√r (3.17)

This means that the binding energy is EB = − κ2

2m , where κ = m2V 20 . In order to �nd

the bound state wave function, we �rst observe that two of the equations in (3.14)do not involve Vr and are therefore independent of renormalization. Their solutionsare

C =A−Be2κr

1− e4κr, D =

B −Ae2κr

1− e4κr(3.18)


The second equation from (3.14) yields

B =Ae2κr

2κ√r

[2κ√r +mV0

(e−4κr − 1

)] r→0−→ A (3.19)

which leads to B = C = A/2, such that the bound state wave function takes theform

ψ(x) = limr→0

ψr(x) = Ae−κ|x| (3.20)

As usual, we proceed by considering scattering states of the form

Ψ(x) =

eikx +Re−ikx, x < −rTeikx, x > r

M1eikx +M2e

−ikx, −r < x < r

(3.21)

Since the potential is antisymmetric, switching the sign of the coupling is equivalentto changing the direction of the incoming plane wave. By combining the boundaryand continuity conditions at ± r, one obtains an inhomogeneous system of equations.In terms of the renormalized coupling, it is given byikeikr + mV0eikr

√r

0 −ikeikr ie−ikrk

0 ikeikr − mV0eikr√r

ike−ikr −ikeikr

eikr 0 −eikr −e−ikr

0 −eikr e−ikr eikr

T

R

M1

M2

=

0

ike−ikr + mV0e−ikr√r

0

e−ikr

(3.22)

In the limit3 , this system is solved by

T =ik

ik +m2V 20

=ik

ik + κ, R = − m2V 2

0

ik +m2V 20

= − κ

κ+ ik,

M1 =∞· sign(V0), M2 = −∞· sign(V0),(3.23)

3.1.3 δ′′ interaction

As before, we write the second derivative in a discretized form, obtaining the Schrödingerequation

− 12m

d2ψ(x)dx2

+Wrδ(x+ r)− δ(x) + δ(x− r)

r2ψ(x) = Eψ(x) (3.24)

3Care must be taken at this point. The limit of the matrix in equation (3.22) for r → 0 is singular,but �rst solving and taking the limit afterwards leads to �nite results for T and R. The constantsM1,2 are in�nite in the said limit, but the region where they are relevant is shrunk to a singlepoint.


To simplify calculations, we �rst look for an even bound state of the form

ψ1(x) =

Ae−κ1x, x > r

Beκ1x + Ce−κ1x, 0 < x < r

Be−κ1x + Ceκ1x, −r < x < 0Aeκ1x, x < −r

(3.25)

on which we impose the boundary and continuity conditions. This yields the systeme−κ1r −e−κ1r −e−κ1r

−(

2mWrr2

+ κ1

)e−κ1r −κ1e

rκ1 κ1erκ1

0 4mWrr2

+ 2κ14mWrr2− 2κ1

A

B

C

= 0 (3.26)

whose determinant must vanish. To the lowest order in r, this implies

4κ1

(κ1 −

4m2W 2r

r3

)= 0 (3.27)

After setting Wr = W0r3/2, we get

κ1 = 4m2W 20 (3.28)

As the intermediate region between ± r is shrunk to a single point, we again end upwith the wave function

ψ1(x) = A1e−κ1|x| (3.29)

We may now proceed towards odd wave functions of the form

ψ2(x) =

De−κ2x, x > r

E sinh(κ2x), −r < x < r

−Deκ2x, x < −r(3.30)

Since this ansatz vanishes at the origin, the middle delta function does not a�ect it.The ensuing conditions read(

1 −eκ2r sinh(κ2r)

−2mWrr2− κ2 −κeκ2r cosh(κ2r)

)(D

E

)= 0 (3.31)

and have nontrivial solutions if

0 =mWr − e2κ2r(mWr + r2κ2)

r2−→ −κ2e

2κ2r

(1 +

mWr

r+ κ2r

)(3.32)


If one applies the renormalization from above, this invariably produces κ2 = 0. Uponan even closer inspection, this equation is found to allow no �nite energy, regardlessof the renormalization applied.

We consider scattering solutions

Ψ(x) =

eikx +Re−ikx, x < −rTeikx, x > r

M1eikx +M2e

−ikx, 0 < x < r

M3eikx +M4e

−ikx, −r < x < 0

(3.33)

on which the boundary and continuity conditions constitute an inhomogeneous sys-tem of equations:

Teikr −Reikr −M1e−ikr = 0

M1 +M2 −M3 −M4 = 0

−Reikr +M3e−ikr +M4e

ikr = e−ikr

Teikr(ik − 2mWr

r2

)−M1ike

ikr +M2ike−ikr = 0

M1

(4mWr

r2+ ik

)+M2

(4mWr

r2− ik

)−M3ik +M4ik = 0

Reikr(ik − 2mWr

r2

)+M3ike

−ikr −M4ikeikr = e−ikr

(ik +

2mWr

r2

)(3.34)

If we substitute the renormalized coupling, the solutions in the limit r → 0 read

T =ik

ik + 4m2W 20

=ik

ik + κ1, R = − 4m2W 2

0

ik + 4m2W 20

= − κ1

ik + κ1,

Mi =∞, i = 1, 2, 3, 4(3.35)

3.1.4 Remarks

The point splitting regularization of the three potentials δ, δ′ and δ′′ corresponds to aone-parameter family of self-adjoint extensions. Due to the form of the transmissionand re�ection coe�cients and the fact that the bound state wave functions ψ(x) =√κe−κ|x| are even and satisfy

∂ψ+(0)− ∂ψ−(0) = −2κψ(0) (3.36)

one deduces that

limx↘0

[ψ(x)

∂ψ(x)

]=

(1 0

c 1

)limx↗0

[ψ(x)

∂ψ(x)

](3.37)


where according to eqs. (3.5), (3.16) and (3.28)

δ potential : c = 2mV0 ∈ R (3.38)

δ′ potential : c = −2m2V 20 < 0 (3.39)

δ′′ potential : c = −8m2W 20 < 0 (3.40)

Hence, c may be an arbitrary real number and each self-adjoint extensions from thisone-parameter family can be approximated by at least one of the three point-splitpotentials.

All the formal expressions presented in this section lead to similar interactions,with δ being the most general one. This is surprising and contradicting the intuitiveidea that the 'internal structure' of a point-split potential may be preserved in thelimit r → 0 if one applies a suitable renormalization, leading to a richer class ofcontact interactions.

3.2 Dimensional regularization

We attempt to solve the Schrödinger equation

− 12m

∆ψ(~x) + V (~x)ψ(~x) = Eψ(~x) (3.41)

in terms of momentum space wave functions

φ(~p) =1

(2π)D/2

∫dDxψ(~x)e−i~p · ~x (3.42)

in arbitrary dimensions D. The Schrödinger equation in momentum space is givenby (

E − p2

2m

)φ(~p) =

1(2π)D/2

∫dDxV (~x)ψ(~x)e−i~p · ~x = F(~p) (3.43)

where F(~p) is the Fourier transform of V (~x)ψ(~x). The previous equation has thesimple solution

φ(~p) =F(~p)

E − p2

2m

(3.44)

3.2.1 Some preliminary calculations

Throughout the following calculations, we will encounter certain divergent integralswhich we will calculate in advance. Denoting by SD−1 the volume of the (D − 1)-


dimensional unit sphere, we have (for positive EB and m)

I1(D,n) =1

(2π)D

∫dDp

|~p|n

EB + p2

2m

=SD−1

(2π)D/2

∞∫0

dp pn+D−1 1

EB + p2

2m

=SD−1

(2π)D

∞∫0

dp pn+D−1

∞∫0

dt exp[−(EB +

p2

2m

)t

]

=SD−1

(2π)D

∞∫0

dt e−EBt

∞∫0

dp pn+D−1e−t

2mp2

=SD−1(2m)

D+n2

2(2π)DΓ(D + n

2

) ∞∫0

dte−EBt

tD+n

2

(3.45)

We note that the initial integral is infrared divergent for n+D ≤ 0 and ultravioletdivergent for n + D ≥ 2, so the expression is really only de�ned for n + D ∈ (0, 2).Ignoring this fact, one can switch integrals and replace the last integral by theanalytic continuation of the Γ function to the whole complex plane except for theorigin and the negative integers, where Γ has simple poles. Finally, one obtains

I1(D,n) =(2mEB)

D+n2

2DEBπ1−D

2∣∣Γ(D2 ) sin(D+n

2 π)∣∣ (3.46)

Similarly, we can derive an expression for the following integral

I2(D,n) =∫dDp

|~p|n(EB + p2

2m

)2 = SD−1

∞∫0

dp pn+D−1

(EB +

p2

2m

)−2

= SD−1

∞∫0

dp pn+D−1

∞∫0

dt exp

[−(EB +

p2

2m

)2

t

]

= SD−1

∞∫0

dt e−E2Bt

∞∫0

dp pn+D−1 exp[−(EBp

2

m+

p4

4m2

)t

](3.47)

The last integral over p evaluates to hypergeometric functions. Finally, one obtains

I2(D,n) =(2mEB)

D+n2 π1+D

2

E2B

∣∣∣∣∣ D+n2 − 1

Γ(D2

)sin(D+n

2 π)∣∣∣∣∣ (3.48)


3.2.2 δ interaction

For V (~x) = V0δ(~x) we have4

F(~p) =1

(2π)D/2

∫dDxV0δ(~x)ψ(~x)e−i~p · ~x =

V0ψ(0)(2π)D/2

(3.49)

in terms of which the bound state wave function with E = −EB is

φ(~p) = − 1(2π)D/2

V0ψ(0)

EB + p2

2m

(3.50)

ψ(0) is determined by normalization provided that the binding energy EB is known.The latter can be determined by a consistency condition

ψ(0) =1

(2π)D/2

∫dDp φ(~p) =

−V0

(2π)D

∫dDp

ψ(0)

EB + p2

2m

⇐⇒ −V0I1(D, 0) = 1

(3.51)

and thus, using the result from equation (3.46),

E1−D

2B =

−V0(2m)D2 π1−D

2

2D∣∣Γ(D2 ) sin

(D2 π)∣∣ (3.52)

This equation can be solved in one and three dimensions:

D = 1 : EB =mV 2

0

2, V0 < 0 (3.53)

D = 3 : EB =2π2

m3V 20

, V0 < 0 (3.54)

In two dimensions, the binding energy is not calculable from these equations 5 .In higher even dimensions, in order to avoid the in�nite right-hand side of equa-tion (3.52), one uses D → D(1 + ε) and absorbs the in�nity by zero renormalizationof the coupling V0 → V0ε. In odd dimensions, renormalization is not required.

By virtue of equation (3.48), the normalization condition of the wave function(3.44) can be rewritten as

1 =∫dDp

∣∣∣∣∣−V0ψ(0)(2π)D/2

1

EB + p2

2m

∣∣∣∣∣2

=∣∣∣∣ V0ψ(0)(2π)D/2

∣∣∣∣2I2(D, 0) (3.55)

4 In dimensions D ≥ 2, it is possible that ψ(0) is divergent, yet ψ ∈ L1(RD) ∩ L2(RD) (seeequations (3.61) and (3.62)). This problem is discussed in Section 3.2.4. What is needed up toequation (3.59) is that F(~p) is �nite and constant.

5The two-dimensional delta potential has some delicate properties which are linked to dimensionaltransmutation and asymptotic freedom. These issues are covered e.g. in [5].


Of the many solutions, we choose the real positive one

− V0ψ(0)(2π)D/2

=1√πEB(2πmEB)−D/4

∣∣∣∣sin(D2 π)

Γ(D

2− 1)∣∣∣∣1/2 (3.56)

Thus, from equation (3.50) we obtain the following normalized momentum spacewave functions:

D = 1 : φ1(p) =

√2π

(2mEB)3/41

2mEB + p2, p ∈ R (3.57)

D = 2 : φ2(~p) =

√2mEBπ

12mEB + p2

, ~p ∈ R2 (3.58)

D = 3 : φ3(~p) =(2mEB)1/4

π

12mEB + p2

, ~p ∈ R3 (3.59)

Let us now proceed to position space wave functions. In one dimension, the calcu-lation is straightforward:

ψ1(x) =1√2π

∞∫−∞

dp φ1(p)eipx = (2mEB)1/4e−√

2mEB |x| (3.60)

In two dimensions, one obtains

ψ2(~x) =1

2π

∫R2

d2p φ2(~p)ei~p · ~x =1

2π2mEB√

π

(1

2mEB

)1/2∞∫−∞

∞∫−∞

dpx dpy eipxx+ipyy

2mEB + p2x + p2

y

=1

2π

√2mEBπ

∞∫−∞

∞∫−∞

dpx dpy eipxx+ipyy

∞∫0

dt e−(2mEB+p2x+p2y)t

=1

2π

√2mEBπ

∞∫0

dt e−2mEBt

∞∫−∞

dpx e−tp2xeipxx

∞∫−∞

dpy e−tp2yeipyy

=

√mEB

2π

∞∫0

dte−2mEBt− 1

4t(x2+y2)

t=

√2mEBπ

K0

(√2mEB|~x|

)(3.61)

where K0 denotes the modi�ed Bessel function of the second kind.

Along the same lines, one obtains the wave function in three dimensions

ψ3(~x) =1

(2π)3/2

∫R3

d3p φ3(~p)ei~p · ~x =(mEB2π2

)1/4 e−√

2mEB |~x|

|~x|(3.62)


3.2.3 ∆δ interaction

Next, we consider V (~x) = W0∆δ(~x). By integration by parts and by sweeping asidesome of the mathematical subtleties6 , we obtain

F(~p) =1

(2π)D/2

∫dDxW0∆δ(~x)ψ(~x)e−i~p · ~x

=W0

(2π)D/2

[∆ψ(0)− 2i~p · ~∇ψ(0)− ~p2ψ(0)

] (3.63)

and for the bound state wave function we have

φ(~p) =−W0

(2π)D/2∆ψ(0)− 2i~p · ~∇ψ(0)− ~p2ψ(0)

EB + p2

2m

(3.64)

Since φ(~p)→ 2mW0ψ(0)

(2π)D/2 as |~p| → ∞, one concludes that ψ(0) = 0 in order to obtain anormalizable wave function. To determine the binding energy, consistency conditionshave to be invoked. They read

∂kψ(0) =1

(2π)D/2

∫dDp ipkφ(~p)

= − W0

(2π)D

∫dDp

ipk∆ψ(0) +∑D

l=1 2pkpl∂lψ(0)

EB + p2

2m

(3.65)

∆ψ(0) =1

(2π)D/2

∫dDp (−~p2)φ(~p)

= − W0

(2π)D

∫dDp

−~p2∆ψ(0) +∑D

l=1 2i~p2pl∂lψ(0)

EB + p2

2m

(3.66)

Integrals involving odd functions such as pl, pkpl, k 6= l or p2pl vanish. By usingequation (3.46), the remaining terms may be summarized as(

2W0

DI1(D, 2) + 1

)~∇ψ(0) = 0 (3.67)

(W0I1(D, 2)− 1)∆ψ(0) = 0 (3.68)

Thus, if W0 > 0, then ~∇ψ(0) = 0 (since I1(D,n) is strictly positive, see equa-tion (3.45)), corresponding to even wave functions

φ0(~p) =−W0

(2π)D/2∆ψ(0)

EB + p2

2m

(3.69)

6As before, the reader is referred to Section 3.2.4, where such issues are discussed properly.


which are of the same form as those from Section 3.2.2 and normalized accordingly.They exist if

W0I1(D, 2)− 1 = 0 (3.70)

As shown in Section 3.2.2, in even dimensions the singularity is avoided with D →D(1 + ε) and zero renormalization of the coupling W0 →W0ε. This leads to 7

D = 1 : EB =1

2m3W 20

, W0 > 0 (3.71)

D = 2 : EB =π

m2W0, W0 > 0 (3.72)

D = 3 : EB =π2/3

21/3m5/3W2/30

, W0 > 0 (3.73)

If on the other hand W0 < 0, then ∆ψ(0) = 0, which implies

φ(~p) =W0

(2π)D/22i~p · ~∇ψ(0)

EB + p2

2m

(3.74)

where ~v = 2i~∇ψ(0) ∈ CD is an arbitrary vector. In principle, it is possible to chooseD mutually orthogonal wave functions

φi(~p) =AW0

(2π)D/2pi

EB + p2

2m

, i = 1, 2, . . . , D (3.75)

where A is a normalization constant determined by the equation

1 =∫dDp

∣∣∣∣∣ AW0

(2π)D/2pi

EB + p2

2m

∣∣∣∣∣2

=∣∣∣∣ AW0

(2π)D/2

∣∣∣∣2 I2(D, 2)D

(3.76)

Although we derived I2(D,n) for arbitrary dimensions in equation (3.48), the wavefunction should still be in L2(RD). Therefore, (3.75) makes sense only in one dimen-sion. The necessary condition for such solutions (see equation (3.68)) reads

2W0

DI1(D, 2) + 1 = 0 (3.77)

Thus, in one dimension one obtains an odd wave function

D = 1 : φ1(p) =(8mE?B)1/4√

π

p

2mE?B + p2, p ∈ R (3.78)

E?B =1

8m3W 20

, W0 < 0 (3.79)

7Negative and complex roots of equation (3.70) are discarded, although they occur very frequentlyin higher dimensions.


which, in position space takes the form

D = 1 : ψ1(x) = (2mE?B)1/4sign(x)e−√

2mE?B |x|, x ∈ R (3.80)

3.2.4 Remarks

Of the two regularizations presented in this chapter, dimensional regularization isperhaps the more problematic and least understood. The procedure is plagued byseveral inconsistencies.

On one hand, the results obtained in Section 3.2.2 and Section 3.2.3 and themathematical treatment from Chapter 2 are compatible. Dimensional regularizationis able to reproduce systems with a single even or odd bound state in one dimension,and the one-parameter families in two and three dimensions. However, the fact thatthe calculated binding energies sometimes decrease with increasing coupling strengthis somewhat disturbing. It might be taken as an indication that by extending theconcept of dimension to the complex plane, certain quantities change their meaningcompletely. In D ≥ 4, we know that contact interactions cannot be de�ned usingself-adjoint extensions. Thus we suspect that if it is possible to de�ne them at all,they must look quite unlike their lower-dimensional counterparts. Presently, thefact that dimensional regularization can be carried to arbitrary dimensions withoutchanging the characteristics of the interaction is not understood.

Let us �rst have a look at the δ potential, which is the less delicate one. Apartfrom the calculation of binding energies, the whole derivation depends on the solecondition that F(~p) be a �nite constant restricted in some way which determines theenergy scale. This requirement is not full�lled by the potential δ(~x) since the positionspace wave functions may exhibit logarithmic or stronger divergences at the origin,thus making the distribution ill-de�ned. A cure for this problem should provide fora consistent translation of the singular potential to momentum space representation,as well as for a credible condition on the grounds of which binding energies canbe calculated. As suggested e.g. in [9], p.46, this can be achieved by a regulatedpotential which spreads the singularity over circles (or spheres) of decreasing radius.

The ∆δ potential, despite its shortcomings, has the bene�t of being the only reg-ularization of contact interactions presented in this work producing a single boundstate of odd parity in one dimension. This result is consistent with the four-parameterfamily of self-adjoint extensions in one dimension. The weakness of the present regu-larization is its lack of physical meaning. First, one is forced to discard one degree offreedom (in equation (3.64)), but nevertheless, one obtains complex binding energieswhich also have to be discarded in dimensions greater than one. Therefore, withoutthese 'surgical interventions', V ∼∆δ simply yields a non-self-adjoint energy opera-tor, which is not a reliable model of quantum mechanical systems. The observationthat dimensional regularization can fail in a non-perturbative theory is quite oftenmade, and discussed e.g. in [5].

Bibliography

[1] Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models inQuantum Mechanics. Texts and Monographs in Physics, Springer-Verlag, NewYork (1988).

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Point Interactions in Nonrelativistic Quantum Mechanics

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