Theoretical and Mathematical Physics, 193(3): 1761–1782 (2017)

SCALARIZATION OF STATIONARY SEMICLASSICAL PROBLEMS

FOR SYSTEMS OF EQUATIONS AND ITS APPLICATION IN

PLASMA PHYSICS

A. Yu. Anikin,∗† S. Yu. Dobrokhotov,∗ A. I. Klevin,∗ and B. Tirozzi‡

We propose a method for determining asymptotic solutions of stationary problems for pencils of differential

(and pseudodifferential) operators whose symbol is a self-adjoint matrix. We show that in the case of

constant multiplicity, the problem of constructing asymptotic solutions corresponding to a distinguished

eigenvalue (called an effective Hamiltonian, term, or mode) reduces to studying objects related only to

the determinant of the principal matrix symbol and the eigenvector corresponding to a given (numerical)

value of this effective Hamiltonian. As an example, we show that stationary solutions can be effectively

calculated in the problem of plasma motion in a tokamak.

Keywords: spectrum, semiclassical asymptotic behavior, plasma equation, tokamak

DOI: 10.1134/S0040577917120042

1. Introduction

The Maslov canonical operator [1] suggests an algorithm for calculating subsequences of asymptoticeigenvalues and asymptotic eigenfunctions (spectral series) for matrix differential and pseudodifferentialoperators (i.e., operators with matrix-valued symbols). The algorithm is very effective for many impor-tant operators encountered in physical problems, but its practical realization can be rather difficult incomplicated concrete situations where it is required to determine functions depending on coordinate andmomentum variables, i.e., the eigenvalues (called modes or effective Hamiltonians) and eigenvectors of theprincipal matrix symbol. The problem of determining the eigenvalues (eigenfunctions) can be easily solvedif the matrix dimension is two. But if the dimension is greater, then the explicit formulas, if any, can bevery cumbersome (like the Cardano formula), and their practical use is rather doubtful. Here, we proposeseveral approaches that allow overcoming such difficulties.

We show that to solve stationary problems, we need not consider the Hamiltonian system relatedto the mode, i.e., determine the Lagrangian manifolds and Maslov indices, solve the transport equation,etc. Instead, it suffices to study a system whose Hamiltonian is the determinant of the principal symbol(which, in contrast to the mode, can be calculated analytically for any dimension of the matrix). We

∗Ishlinskii Institute for Problems in Mechanics, RAS, Moscow, Russia; Moscow Institute of Physics and Tech-

nology, Dolgoprudny, Moscow Oblast, Russia, e-mail: anikin83@inbox.ru, dobr@ipmnet.ru.†Bauman Moscow State Technical University, Moscow, Russia.‡ENEA Centro Ricerch di Frascati, Frascati (Roma), Italy, e-mail: brunellotirozzi@gmail.com.

This research is supported by the Russian Foundation for Basic Research (Grant Nos. 14-01-00521 and 16-31-

00339).

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 193, No. 3, pp. 409–433, December, 2017.

Original article submitted December 16, 2016; revised June 22, 2017.

0040-5779/17/1933-1761 c© 2017 Pleiades Publishing, Ltd. 1761

call such a reduction of a vector problem to a scalar problem scalarization. This method is based on thesemiclassical analogue of the Maupertuis–Jacobi principle (see [2]). Here, we consider only the case ofconstant multiplicity.

An important fact is that the semiclassical approximation of stationary problems is based on thetrajectories of the classical Hamiltonian system for which the effective Hamiltonian (energy or squaredfrequency) is a global constant, and precisely this allows replacing an eigenvalue of the matrix symbol withits determinant in asymptotic constructions. Similar arguments were used in [3] for differential equationson a complex manifold. The determinant was used as the Hamiltonian in [4].

Our work is motivated by the stationary problem for the linearized system of equations describing themotion of cold plasma in a tokamak [5]. In this problem, the operator symbol is a 12×12 matrix, and it ishence impossible to calculate its eigenvalues directly. We show that such a spectral problem is reducible toa problem for a pencil of operators, where the symbol is already a 3×3 matrix.

The problem for a pencil of operators is the problem H(x, p, E)Ψ(E) = 0, where the operator dependson a parameter. This is a standard spectral problem in the particular case H = G(x, p) − E. We seekasymptotic solutions of this problem, i.e., a number E and a function Ψ (with the normalization condition‖Ψ‖L2 = 1) such that ‖ HΨ‖L2 = O(hβ), where β > 0. The numbers E are called asymptotic eigenvalues,and the set of asymptotic eigenvalues in an interval is called a spectral series. The functions Ψ(E) are calledasymptotic solutions or quasimodes.

We note that problems for operator pencils are well known in the theory of partial differential equationsand in fluid mechanics. But the usual method for studying such problems is opposite in a sense to thatused here, namely, a more complicated problem of studying the pencil is reduced to the standard spectralproblem [6].

In the plasma problem cited above, we propose an explicit formula for the determinant of the principalsymbol. It is very cumbersome and leads to a nonintegrable Hamiltonian system in the typical case.Additional efforts (using averaging theory, normal forms near a single torus, etc.) are therefore required toconstruct spectral series using Lagrangian tori.

A simpler problem is to construct spectral series corresponding to asymptotic solutions localized near astable equilibrium or an orbitally stable closed (elliptic) curve. The case of a neighborhood of an equilibrium(the so-called lower energy levels) is well studied and does not encounter great difficulties. The solutionscorresponding to a closed elliptic curve are constructed in the framework of the theory of the Maslov complexgerm [7] (also see [8]). We show that the spectral series and the asymptotic eigenfunctions constructed usingthe complex germ for a mode can be obtained from the complex germ for the determinant. We thus presentan effective algorithm for determining such spectral series numerically.

This paper is organized as follows. In Sec. 2, we recall the well-known semiclassical formulas in scalarand vector stationary problems for operator pencils. In Sec. 3, we scalarize the solutions related to Liouvilletori. In Sec. 4, we recall the theory of the complex germ over a closed curve and perform the scalarization insuch a case. In Sec. 5, we calculate the determinant of the principal symbol for the equation of a linearizedplasma. We also consider a specific example of a system with closed trajectories generating the spectralseries, which can be calculated effectively.

2. Asymptotic solutions of stationary problems for operatorpencils

2.1. Asymptotic solution in the scalar case. We first consider the equation for a pencil of dif-ferential or pseudodifferential operators

H(E)Ψ(E) ≡ H(2x,

1

p, E, h)Ψ(x, E, h) = 0

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with the symbol H(x, p, E, h) = H0(x, p, E) + hH1(x, p, E) + . . . (if H(x, p, E, h) is a polynomial in p, thenH(E) is a differential operator; the numbers above symbols denote the order of action of the operators [9]).We assume that the function H0(x, p, E) is real and related to a Hamiltonian vector field vH0 (dependingon a parameter E) with the phase flow gt

H0in the standard symplectic space associated with the system of

equations

p = −∂H0

∂x, x =

∂H0

∂p.

We setΣE = {(x, p) : H0(x, p, E) = 0} ⊂ R

2nx,p, ˜Σ =

⋃

|E−E0|<ε

ΣE ,

and Σ = ˜Σ ∩ U , where U ⊂ R2n is a given open set. We assume that

(A1) (H0)′E �= 0 everywhere in Σ and

(A2) Hamiltonian systems with the Hamiltonians H0 = H0(E) for |E − E0| < ε are completely integrableon Σ.

Remark 1. These assumptions can be weakened by assuming that the system corresponding to thevalue of the parameter E at the energy level ΣE ∩ U is integrable.

We now construct the action–angle variables in the domain Σ for all Hamiltonian systems H0(E) (for|E − E0| < ε), namely, p = P (I, ϑ, E) and x = X(I, ϑ, E). Let Λn(I, E) ⊂ Σ be invariant tori for thesystem H0(E) corresponding to their values of I. We assume that the values Iν

j = h(νj + mj/4) and E

correspond to tori satisfying the Bohr–Sommerfeld quantization rule. Here, ν = (ν1, . . . , νn) ∈ Zn+, and mj

is the Maslov index of the cycle corresponding to ϑj . For each set Iνj , we choose a value Eν = E(Iν) (unique

by assumption A1) such that H0(Iν , Eν) = 0. In other words, we fix a torus Λ = Λ(Iν , Eν) ⊂ ΣEν .We assume that the functions Aλ : Λ → R and λ satisfy the transport equation

dAλ

dt+(

−12

tr∂2H0

∂x ∂p+ iH1 − λ

∂H0

∂E

)∣

∣

∣

∣

E=Eν

Aλ = 0, (1)

where t is the Hamiltonian time, i.e., d/dt = {H0, · } ≡ adH0 , and∫

Λ

|Aλ|2 dϑ1 · · · dϑn = 1.

We construct a family of functions ψν : Rnx → R using the Maslov canonical operator: ψν = KΛn(Iν)Aλ.

Here, KΛn(Iν) : C∞(Λ) → C∞(Rnx) is a canonical operator constructed on the Lagrangian manifold Λn(Iν).

We here assume that a point (whose contribution is a constant unimodular factor) is fixed on Λn(Iν) andan invariant measure dμ (equal to dϑ1 · · · dϑn up to a factor) is chosen.

We now recall the form of the canonical operator in a nonsingular chart. Let χα ∈ C∞(Λn(Iν)) be afunction compactly supported in a chart Λα ⊂ Λ, and let Λα project diffeomorphically on R

nx . Then

KΛn(Iν)[Aχα] =A(ϑ(x))χα(ϑ(x))√

∣

∣∂X∂ϑ (ϑ(x))

∣

∣

eiS(ϑ(x))/h−imπ/2. (2)

Here, S(ϑ) =∫ ϑ

θ0p dx, ϑ = ϑ(x) can be determined from X(ϑ(x)) = x, and m is the Maslov index of a path

connecting the points with the coordinates ϑ0 and ϑ on the torus Λ (see [1]).We set

E(Iν) = E(Iν) + hλ(Iν).

We then have (see [1])∥

∥ H(E(Iν))ψν∥

∥

L2(Rn)= O(h2).

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2.2. Asymptotic solution in the vector case. We consider the spectral problem for a pencil ofoperators whose symbol is a self-adjoint N×N matrix

H(E)Ψ = 0, H = H0 + hH1 + . . . .

We introduce the notation ‖a‖ = |a1|2 + · · · + |aN |2 and 〈a, b〉 = a1b1 + · · · + aNbN and let a bar overa symbol denote complex conjugation.

The algorithm (see [1] and [10]) for determining the asymptotic solution Ψ = Ψ(h) and E = E(h) isas follows:

1. Obtain the eigenvalues and eigenvectors of the principal symbol (modes or effective Hamiltonians):

H0χr = Hrχr, ‖χr‖ = 1, r = 1, . . . , N.

2. Choose a mode Hr and calculate the scalar function

Lr = 〈χr,H1χr〉 − i

⟨

χr,dχr

dt

⟩

− i

⟨

χr,

n∑

j=1

[

∂H0

∂pj− ∂Hr

∂pjI]

∂χr

∂xj

⟩

, (3)

where d/dt = adHr .

3. Obtain an asymptotic solution of the scalar problem

( Hr(E) + hLr(E))

ψr(E) = 0, (4)

which determines the spectral series E = E(h).

4. Calculate an asymptotic solution of the initial value problem using the formula

Ψr = χrψr, χr = χr(2x,

1

p).

To realize this algorithm in practice, we must therefore have explicit expressions for the eigenvalues Hr ofthe matrix H0. Such formulas can be easily obtained in problems with matrices of dimension two. Theformulas are well known for some classical systems of equations (Maxwell and Dirac equations, elasticityequations, linearized Navier–Stokes equations), but they are too cumbersome or do not exist at all for manyother equations of larger dimensions. For example, this is the situation with the system of linearized plasmaequations considered here. We propose a trick for avoiding the encountered difficulty. Namely, we write theasymptotic solution in terms of the system with not the Hamiltonian H but the Hamiltonian H = detH0.

3. Scalarization

3.1. Maupertuis–Jacobi principle. We show how to reduce the Hamiltonian system vHr to asystem whose Hamiltonian is the determinant of the matrix H0. The following simple argument, whichis naturally called the Maupertuis–Jacobi principle (see [11]), plays the key role here. If H, G : R

2n → R

are two functions in the symplectic space and Σ ⊂ R2n is a hypersurface on which the functions H and

G are constant, then the vector fields vH and vG are parallel everywhere on Σ (and the solutions of theHamiltonian systems hence coincide up to a change of time).

We fix one mode, namely, Hr(x, p, E). Let Λn be a Lagrangian manifold invariant under gtHr

(for agiven value of E) and such that Hr

∣

∣

Λn = 0. We assume the following.

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Assumption 1. We have Hj

∣

∣

Σr�= 0 for all j �= r.

The mode Hr thus has a constant multiplicity in the domain of interest.We consider the determinant of the principal symbol

H(E) = detH0(E) ≡ H1(E) · · ·Hn(E)

and write the characteristic polynomial of the matrix H0 as

æ(E, μ) = det(H0 − μI) = (−1)nn∏

j=1

(μ − Hr(E)).

Everywhere on Σr, we then have∏

j �=r

Hj =∂æ∂μ

∣

∣

∣

∣

μ=0

= −I(E),

where I(E) is the sum of (n−1)th-order principal minors of the matrix H0 (which is nonzero by Assump-tion 1).

Therefore, we have

Hr(E) = Fr(p, x, E)H, Fr

∣

∣

Σr= − 1

I(E)�= 0.

We assume that ‖χr‖ = 1. The following statement then holds.

Theorem 1. Let Assumption 1 hold. Then the transport equation for Λn for the operator in the

left-hand side of (4) becomes

dAλ

ds+[

i

Fr〈χr,H1χr〉 +

⟨

χr,dχr

ds

⟩

− 12Fr

⟨

χr,

n∑

j=1

∂2H0

∂pj ∂xjχr

⟩

+

+ i Im⟨

χr,

n∑

j=1

[

1F

∂H0

∂pj− ∂H

∂pjI]

∂χr

∂xj

⟩

− λ∂H∂E

]

Aλ = 0, (5)

where d/ds = adH.

Proof. The proof is given in Appendix A.

3.2. Determination of χr and ∂χr/∂xj. Because Λn ⊂ Σr, it is easy to see that all objects inEq. (5) are either known or can be calculated by only arithmetic operations. We explain how to calculate theleast trivial objects, i.e., the eigenvector χr and its derivative ∂χr/∂xj . The eigenvector can be determinedfrom the equation H0χr = 0 everywhere on Σr. To obtain the derivative, we differentiate the identityH0χr = Hrχr with respect to xj and obtain an inhomogeneous linear equation for ∂χr/∂xj on Σr,

H0∂χr

∂xj= F

∂H∂xj

χr −∂H0

∂xjχr.

We note that the solution of such an equation is not unique on Σr but is determined up to the vector χr

multiplied by a constant. It follows from identity (32) that all such solutions (even if they are not in factequal to ∂χr/∂xj) contribute the same to Eq. (5) and can be substituted there.

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3.3. Invariant measure. After determining the solution Aλ of Eq. (5), there is still one obstacle inthe way of obtaining the asymptotic form of ψr(E) (problem (4)). Namely, we must know the invariantmeasure of the system vHr . Let ϑ = (ϑ1, . . . , ϑn) be the angle variables on the torus Λn for the system vH(with the dynamics ϑ = ω, where ω is a constant frequency vector). The dynamics in the variables ϑ is thengiven by the equations ϑ = ωF for the system vHr . The measure dμ = dϑ1 · · · dϑn/F is hence invariant forthe system vHr and can be used to construct the canonical operator.

Therefore, we can write the asymptotic solution in terms of the Hamiltonian system vH. We have thusproved the following theorem.

Theorem 2. Let the Hamiltonian system vH be completely integrable and have invariant Liouville

tori in a domain of the phase space. Let I and ϑ be the action–angle variables in this domain. Let a torus

Λ = Λ(I, E0) corresponding to I = Iν satisfy the Bohr–Sommerfeld quantization rule Iνj = h(νj + mj/4).

Let the torus Λ be on the surface H(E0) = 0 and the function Hr(E0) be zero on it while all other functions

Hj(E0) are nonzero. Let χr(p, x) be a vector function such that ‖χr‖ = 1 (here ‖ · ‖ is the Euclidean norm

of a vector), and let H0(E0)χr = 0 in a neighborhood of the torus Λ on the surface H(E0) = 0. Then there

is an asymptotic solution of the problem H(E0, h)Ψ(h) = 0 of the form

E = E0 + λh, Ψ = χrKΛAλ.

Here, KΛ : C∞(Λ) → C∞(Rnx) is the canonical operator constructed on the Lagrangian manifold Λ, where

the invariant measure dϑ1 · · · dϑn/F is chosen, and the function Aλ and the number λ form a solution of

Eq. (5).

Remark 2. The canonical operator in a nonsingular chart becomes

KΛ[χαA] =A(ϑ(x))χα(ϑ(x))

√

F (ϑ(x))√

|∂X/∂ϑ|eiS(ϑ(x))/h−mπi/2

(cf. formula (2)).

4. Scalarization for the solution localized near a closed curve

4.1. Complex germ over a closed curve. The Maslov complex germ theory allows relating thespectral series to a family of isotropic invariant tori Λk (k < n). The simplest examples of such manifoldsare a point (k = 0) and a closed curve (k = 1). The case k = 0 corresponds to the well-studied lower energylevels (harmonic oscillator approximation). Here, we are interested in the simplest nontrivial case k = 1.

We now recall the construction of a complex germ in the case k = 1 (the general case can be foundin [7]).

Let R2np,x be a symplectic space with the standard form ω = dp ∧ dx. A smooth Hamiltonian function

H : R2n → R determines a vector field with the phase flow gt

H : R2n → R

2n in this space. We assume thatthis Hamiltonian system has an orbitally stable T -periodic solution γ:

x = X0(t), p = P0(t).

We further assume that the following nondegeneracy and regularity condition is satisfied.

Assumption 2. 1. The multipliers of the solution γ have the forms

μ1 = μn+1 = 1, μk, μk+n = e±iβk for k = 2, . . . , n,

where βk > 0. This means that the curve γ is orbitally stable in the linear approximation.

2. The curve γ in the phase space can be projected diffeomorphically on the configuration space Rnx .

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For k �= 1, n+1, we let vk =(

Bk

Ck

)

denote Floquet solutions of the variational system near the solution γ:

B = −HxpB − HxxC, C = HppB + HpxC, (6)

i.e., vk(t + T ) = μkvk(t) under the assumption that vk+n = vk.We choose n solutions of system (6) as ξ1 =

(

P0(t)

X0(t)

)

, and for k = 2, . . . , n, we set either ξk = vk orξk = vk+n such that the matrix

(

Im(ω(ξj , ξk )))

∣

∣

∣

n

j,k=1≡(

Im(BTj Ck − CT

k Bj ))

∣

∣

∣

n

j,k=1(7)

is positive definite. Such a choice is always possible because of the following. This matrix is diagonalbecause the vectors vk and vj are skew orthogonal if μkμj �= 1. Moreover, it is easy to see that the quantityω(vk, vk) is purely imaginary, and we can hence satisfy (7) by suitably choosing ξk. Such a condition iscalled dissipativity in the general complex germ theory.

We let( BC)

denote a matrix of n columns ξ1, . . . , ξn and 2n rows. The vectors ξ1, . . . , ξn generate afamily of complex n-dimensional planes constructed over the points of a closed trajectory γ. The vectorbundle thus obtained is called an (isotropic) one-dimensional manifold with a complex germ.

Such an object generates a spectral series. For this, it is necessary to determine the phase and theamplitude. Point 2 in Assumption 2 implies [7] that detC �= 0 everywhere on γ. Hence, there are no focalpoints in our case. (Our further results are generalizable to a more general case with focal points, whichonly technically complicates the statements.)

In a small neighborhood of the projection of γ on Rnx , we define a function t = t(x) by the rule

〈x−X0(t), X0(t)〉 = 0 (i.e., t corresponds to a point on the curve γ such that the normal to γ at this pointcontains x). We can define a function S0 = S0(t) on the curve γ such that dS0 = p dx. We introduce thephase

S(x) = S0(t) + 〈P0(t), x − X0(t)〉 +12〈BC−1(x − X0(t)), x − X0(t)〉

∣

∣

t=t(x). (8)

The amplitude Aλ : γ → R can be determined from the transport equation, which has precisely thesame form (1) as in the case of a complete dimension.

These objects were generally defined not on the curve γ but only on its universal covering. To attaina good sewing consistency between these objects, we must impose the quantization condition

12πh

∮

γ

p dx =1

2πT

n−1∑

j=1

bj(γE)(

νj +12

)

+ νn, (9)

where ν1, . . . , νn−1 are fixed nonnegative integers (independent of h) and the number νn takes values ∼ 1/h.Simply speaking, the number νn corresponds to the Bohr–Sommerfeld-type quantization for periodic

trajectories of a one-dimensional operator. The numbers ν1, . . . , νn−1 correspond to harmonic-oscillator-type states in the direction transverse to γE . The formulas are significantly simpler in the case of the“ground” state, i.e., for ν1 = · · · = νn−1 = 0. We consider precisely this case below.

The asymptotic form of the eigenvalues of the initial operator is

E(h) = H(γ) + hλ + O(h2),

and that of the eigenfunctions becomes

ψ =1

√

det C(t(x))Aλ(t(x))eiS(x)/h

for ν1 = · · · = νn−1 = 0. If ν �= 0, then the formulas are more complicated and contain analogues of creationoperators (see [7]).

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4.2. Scalarization. As in Sec. 3.1, we choose a mode Hr = FrH (we omit the index r in whatfollows). We show how to construct a stationary solution of the initial system if a periodic solution with acomplex germ is known for the system with the Hamiltonian H.

Let γ be a Ts-periodic solution of the system vH such that Assumption 2 and quantization condition (9)are satisfied.

We assume that z = (p, x) and the Hamiltonians H and H determine the systems

dz

dt= vH = FvH,

dz

ds= vH, H = FH, ds = Fdt.

Let z = ZH(s) be a periodic solution for vH with the period Ts and z = ZH(t) = Z(H(s(t)) be a periodicsolution for vH with the period Tt. We consider the two variational systems

dξ

dt= ΩH(t)ξ, ΩH(t) =

(

−Hxp −Hxx

Hpx Hpp

)

∣

∣

∣

∣

z=ZH (t)

, (10)

dη

ds= ΩHη, ΩH(s) =

(

−Hxp −Hxx

Hpx Hpp

)

∣

∣

∣

∣

z=ZH(s)

. (11)

Lemma 1. Let η(s) be a Floquet solution (η(s + Ts) = μη(s)) of system (11) such that dH(η) = 0.

Let a scalar function r = r(t) be a solution of the problem

dr

ds= r

{F,H}F

+dF (η)

F, r(s + Ts) = μr(s). (12)

Then the function

ξ(t) = η(s(t)) + r(s(t))vH(ZH(s(t))) (13)

is a Floquet solution ξ(t + Tt) = μξ(t) of system (10).

Remark 3. Of course, we can write an explicit formula for r(t), but it is rather cumbersome and notinteresting for us.

Proof. The proof is given in Appendix B.

For the system vH, we construct a matrix(BC)

the same as in Sec. 4.1 (its first column is vH(ZH(s))).We construct the matrix

(

BC

)

, where the first columns is vH(ZH(t)) and the other columns are obtainedfrom the corresponding columns of the matrix

( BC)

by applying Lemma 1. Speaking somewhat loosely, wesay that the matrices

( BC)

and(

BC

)

are closed curves with a complex germ of the systems vH and vH . Thenext statement then follows directly from Lemma 1.

Corollary 1. 1. If quantization conditions (9) are satisfied for a closed curve with a complex germ of

the system vH, then they are also satisfied for a closed curve with a complex germ of the system vH .

2. We have BC−1 = BC−1, and the phases S(x) determined by the closed curves with complex germs

vH and vH hence coincide.

3. The Jacobians of the two closed curves with complex germs are equal to each other:

det C(s(x)) = detC(t(x)).

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Proof. Point 1 follows because the multipliers are the same for the two variational systems. Points 2and 3 follow because the matrices B and C are obtained from the matrices B and C by elementary trans-formations of the columns.

We write the stationary solution explicitly. Let γ be a trajectory of the system dz/ds = vH(z)corresponding to the value E = E0 at the level H = 0. We calculate the functions s(x), S(x), and C(x). Atthe points of the trajectory γ, we calculate the function

G(s) =i

Fr〈χr,H1χr〉 +

⟨

χr,dχr

ds

⟩

− 12Fr

⟨

χr,

n∑

j=1

∂2H0

∂pj∂xjχr

⟩

+

+ i Im⟨

χr,

n∑

j=1

[

1F

∂H0

∂pj− ∂H

∂pjI]

∂χr

∂xj

⟩

and set

λ0 =

∫ Ts

0 G(s) ds∫ Ts

0∂H∂E (s) ds

, Aλ0(s) = exp[ ∫ s

0

(

G(τ) − λ0∂H∂E

(τ))

dτ

]

. (14)

The above argument implies the following statement.

Theorem 3. The asymptotic solution generated by the complex germ over a closed curve corresponds

to the mode Hr and has the form

E = E0 + hλ0, ψ(x) =1

√

det C(s(x))Aλ0(s(x))e

iS(x)/h

in the case of constant multiplicity.

Remark 4. In the semiclassical theory of systems of equations, there arise effects of changing themultiplicity of the characteristics (effects of mode intersection). The effective Hamiltonians correspondingto different modes then coincide on some surfaces in the phase space. For the asymptotic solutions listedabove, it is important that such effects do not arise on the corresponding closed curve Λ1. It is easy to seethat such a case is impossible because some equilibrium points would otherwise appear on the trajectory.

4.3. Explicit formulas in a particular case. We consider a particular case that is interestingin applications related to plasma physics. We assume that the Hamiltonian H is given in a phase spacewith the coordinates x, y, z, px, py, pz, where x, y, z are Cartesian coordinates in the configuration (physical)space.

We introduce toroidal coordinates in the configuration space:

x = Q(r, θ) cosφ, y = Q(r, θ) sin φ, z = r sin θ, (15)

where Q(r, θ) = R0 + r cos θ. The Jacobi matrix becomes

J3 =∂(

x, y, z)

∂(

r, θ, φ) =

⎛

⎜

⎜

⎝

cos θ cosφ −r sin θ cosφ −Q sinφ

cos θ sinφ −r sin θ sin φ Q cosφ

sin θ r cos θ 0

⎞

⎟

⎟

⎠

.

The canonical change of coordinates is defined in the phase space,

dpx ∧ dx + dpy ∧ dy + dpz ∧ dz = dpr ∧ dr + dpθ ∧ dθ + dpφ ∧ dφ,

1769

where⎛

⎜

⎜

⎝

px

py

pz

⎞

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎜

⎜

⎝

cos θ cosφ −1r

sin θ cosφ − 1Q

sin φ

cos θ sin φ −1r

sin θ sin φ1Q

cosφ

sin θ1r

cos θ 0

⎞

⎟

⎟

⎟

⎟

⎟

⎠

⎛

⎜

⎜

⎝

pr

pθ

pφ

⎞

⎟

⎟

⎠

≡ (J−13 )T

⎛

⎜

⎜

⎝

pr

pθ

pφ

⎞

⎟

⎟

⎠

.

The Jacobi matrix has the form

J6 =∂(px, py, pz, x, y, z)∂(pr, pθ, pφ, r, θ, φ)

=

(

(J−13 )T M

0 J3

)

,

where

M =

⎛

⎜

⎜

⎜

⎝

m11 m12 m13

m21 m22 m23

−pθ

r2cos θ cos θpr −

pθ

rsin θ 0

⎞

⎟

⎟

⎟

⎠

,

m11 =pθ

r2sin θ cosφ +

pφ

Q2cos θ sinφ,

m12 = − sin θ cosφpr −pθ

rcos θ cosφ − pφ

Q2r sin θ sin φ,

m13 = − cos θ sinφpr +pθ

rsin θ sinφ − pφ

Qsin φ,

m21 =pθ

r2sin θ sin φ − pφ

Q2cos θ cosφ,

m22 = − sin θ sin φpr −pθ

rcos θ sinφ +

pφ

Q2r sin θ cosφ,

m23 = cos θ cosφpr −pθ

rsin θ cosφ − pφ

Qsin φ.

Assumption 3. 1. The variable pφ is the first integral (cyclic) of the Hamiltonian system, i.e., H =H(r, θ, pr, pθ, pφ, E).

2. For all pφ ∈ (p1φ, p2

φ) and E = E(pφ), a nondegenerate minimum z0 = (r0(pφ), θ0(pφ), pr0 = 0, pθ = 0)exists, where H(z0, pφ, E(pφ)) = 0.

3. We have ∂H/∂E �= 0, ∂H/∂pφ �= 0, and Fr �= 0 at the point z0, pφ, E(pφ).

Therefore, the system has an orbitally stable periodic trajectory generating a stationary solution. Wenow calculate all objects in the asymptotic solution explicitly.

We first note that formulas (14) can be rewritten as

λ0 =G(z0, pφ, E(pφ))

∂H/∂E(z0, pφ, E(pφ)), Aλ0 = 1. (16)

To determine the matrices B and C, we first calculate them in toroidal coordinates:

Btor = (B(1),B(2),B(3)), Ctor = (C(1), C(2), C(3)),

whereB(j)

tor = (B(j)r ,B(j)

θ ,B(j)φ )T, C(j)

tor = (C(j)r , C(j)

θ , C(j)φ )T.

1770

Then B(1) = 0 and C(1) = (0, 0, ∂H/∂φ)T. To find the second and third columns, we must write thevariational system whose normal part has constant coefficients:

d

ds

⎛

⎜

⎜

⎜

⎜

⎝

Br

Bθ

Cr

Cθ

⎞

⎟

⎟

⎟

⎟

⎠

= Ω

⎛

⎜

⎜

⎜

⎜

⎝

Br

Bθ

Cr

Cθ

⎞

⎟

⎟

⎟

⎟

⎠

, Ω =

⎛

⎜

⎜

⎜

⎜

⎝

−Hr pr −Hr pθ−Hr r −Hr θ

−Hθ pr −Hθ pθ−Hθ r −Hθ θ

Hpr pr Hpr pθHpr r Hpr θ

Hpθ pr Hpθ pθHpθ r Hpθ θ

⎞

⎟

⎟

⎟

⎟

⎠

. (17)

Here, the matrix Ω is calculated at the point z0. The Floquet solutions of this system

(Br,Bθ, Cr, Cθ)T(s + Ts) = μ(Br,Bθ, Cr, Cθ

)T(s)

are used to construct the Floquet solutions of the whole system. For this, we must set Bφ ≡ 0, and as Cφ,we must take the solution of the problem

dCφ

ds= Hpφ pφ

Cφ + F(s), Cφ(s + Ts) = μCφ(s),

whereF(s) = Hpφ pr (z0)Br(s) + Hpφ pθ

(z0)Bθ(s) + Hpφ r(z0)Cr(s) + Hpφ θ(z0)Cθ(s).

In explicit form, we have

Cφ(s) =1

μ − eTsHpφ pφ

∫ Ts

0

e(Ts+s−τ)Hpφ pφF(τ) dτ +∫ s

0

e(s−τ)Hpφ pφF(τ) dτ.

The required columns of the matrices B and C are finally obtained by the formula

(Bx,By,Bz, Cx, Cy, Cz)T = J6(Br,Bθ, 0, Cr, Cθ, Cφ)T. (18)

We return to system (17). Let ±βj = ±ibj for j = 1, 2, b1, b2 > 0, be eigenvalues of the matrix Ω. Wemust understand which two eigenvalues and eigenvectors to take to satisfy the dissipativity condition.

Lemma 2. Let a 2n×2n matrix A be real, symmetric, and positive definite. Let the matrix of the

system

η =

(

0 −I

I 0

)

Aη

be diagonalizable and have the eigenvalues ±ibj, where bj > 0, j = 1, . . . , n. Let vj be the eigenvector

corresponding to ibj. Then the solution ηj = vjeibjs satisfies the inequality

Im ω(ηj , ηj) > 0.

Proof. For the matrix of the system, there is a symplectic change of coordinates reducing this matrixto the diagonal form. The general case is therefore reducible to studying the case of dimension n = 1. Wehave

η =

(

−a12 −a22

a11 a12

)

η, a11 > 0, Δ = a11a22 − (a12)2 > 0.

Then λ = i√

Δ, v = (a11, a12 − i√

Δ)T, and ω(v, v) = 2ia11

√Δ.

1771

Lemma 3. 1. Let

v(1) = (v(1)pr

, v(1)pθ

, v(1)r , v

(1)θ )T, v(2) = (v(2)

pr, v(2)

pθ, v(2)

r , v(2)θ )T

be the eigenvectors corresponding to the eigenvalues ib1 and ib2 with positive imaginary parts. Then

Btor(Ctor)−1 =

⎛

⎜

⎝

Bnorm(Cnorm)−1 00

0 0 0

⎞

⎟

⎠,

where

Bnorm =

(

v(1)pr v

(2)pr

v(1)pθ v

(2)pθ

)

, Cnorm =

(

v(1)r v

(2)r

v(1)θ v

(2)θ

)

. (19)

2. We have the relation

BC−1 = (J−13 )T

⎛

⎜

⎜

⎜

⎜

⎜

⎝

Bnorm(Cnorm)−1−pφ cos θ

Qpφr sin θ

Q

−pφ cos θ

Q

pφr sin θ

Q0

⎞

⎟

⎟

⎟

⎟

⎟

⎠

J−13 . (20)

3. In the particular case where the Hessian of H in the variables r, θ, pr, pθ is diagonal and Hrr = ν1,

Hθθ = ν2, Hprpr = ν3, and Hpθpθ= ν4, we have

BC−1 = (J−13 )T

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎝

i

√

ν1

ν30 −pφ cos θ

Q

0 i

√

ν2

ν4

pφr sin θ

Q

−pφ cos θ

Q

pφr sin θ

Q0

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎠

J−13 .

4. The relation

det C(s) = C0ei(b1+b2)s

holds for a constant C0.

Proof. 1. We use the matrices of elementary transformations of the columns and easily obtain

Btor(Ctor)−1 =

⎛

⎜

⎜

⎝

0 eib1sv(1)pr eib1sv

(2)pr

0 eib1sv(1)pθ eib1sv

(2)pθ

0 0 0

⎞

⎟

⎟

⎠

⎛

⎜

⎜

⎜

⎝

0 eib1sv(1)r eib1sv

(2)r

0 eib1sv(1)θ eib1sv

(2)θ

∂H∂pφ

C(1)φ C(2)

φ

⎞

⎟

⎟

⎟

⎠

−1

=

=

⎛

⎜

⎝

0

0Bnorm

0 0 0

⎞

⎟

⎠

⎛

⎜

⎜

⎜

⎝

0

0Cnorm

∂H∂pφ

0 0

⎞

⎟

⎟

⎟

⎠

−1

=

⎛

⎜

⎝

Bnorm(Cnorm)−10

00 0 0

⎞

⎟

⎠.

2. It follows from (18) that B = (J−13 )TBtor + MCtor and C = J3Ctor. Therefore,

BC−1 = (J−13 )TBtorC−1

torJ−13 + MJ−1

3 = (J−13 )T(BtorC−1

tor + JT3 M)J−1

3 .

Taking into account that pr = pθ = 0 at the point z0, we obtain the required statement.3. This assertion can be verified by a direct calculation.4. This equality is obvious.

1772

The lemma readily implies the following statement.

Theorem 4. Let Assumption 3 be satisfied and pφ = p0φ satisfy the quantization condition

p0φ = h

(

k +(b1 + b2)Ts

4π

)

, k ∈ Z.

Let E = E(p0φ) + hλ0, where λ0 can be determined by formula (16), and let

ψ = exp[

i

hS0(s) +

2πs

Ts(kh − p0

φ) + 〈P0(s), x − X0(s)〉 +12〈BC−1(x − X0(s)), x − X0(s)〉

]

,

where BC−1 is determined by (19) and (20) and s = s(x) everywhere by the condition 〈x−X0(s), X0(s)〉 = 0.

Then the pair E, ψ determines the asymptotic eigenvalue and eigenfunction of the stationary problem.

5. An example in plasma physics

5.1. Plasma equation and its linearization. We consider an ionized gas with charges of two types.We assume that the constant charges qi, masses mi, and densities ni, i = 1, 2, are known. The expressionfor the current becomes

J = n1q1v1 + n2q2v2.

We assume that the applied magnetic field B is significantly stronger than the magnetic field generatedby the charge motion. The equation for the electric field is obtained by removing the magnetic field B:

c2∇ ∧ ∇ ∧ E +∂J∂t

+∂2E∂t2

= 0.

Here, ∧ denotes the vector product. We add the Newton equation for charges of two types to obtain thesystem of equations

c2∇ ∧ ∇ ∧ E +∂J∂t

+∂2E∂t2

= 0,

dv1

dt=

q1

m1

(

E +1cv1 ∧ B

)

,

dv2

dt=

q2

m2

(

E +1cv2 ∧ B

)

.

(21)

This system is nonlinear becausedvj

dt=

∂vj

∂t+ 〈vj ,∇〉vj .

We linearize it neglecting the terms 〈vj ,∇〉vj . After linearization, we can rewrite system (21) as

∂2E∂t2

+ c2∇ ∧ ∇ ∧ E +n1q

21

m1

(

E +1cv1 ∧ B

)

+n2q

22

m2

(

E +1cv2 ∧ B

)

= 0,

∂v1

∂t− q1

m1

(

E +1cv1 ∧ B

)

= 0,

∂v2

∂t− q2

m2

(

E +1cv2 ∧ B

)

= 0,

(22)

1773

where B is a known function and E, v1, and v2 are unknown. System (22) is considered inside a solid torusin R

3. We introduce an artificial parameter h, change the variables by setting x = cx′/h and t = t′/h, andwrite t instead of t′ in what follows.

We introduce a new vector function W = −ih(∂E/∂t) and rewrite Eq. (22) as

ih∂E∂t

= −W,

ih∂W∂t

= (−ih∇) ∧ (−ih∇) ∧ E− n1q21

m1

(

E +1cv1 ∧ B

)

− n2q22

m2

(

E +1cv2 ∧ B

)

,

ih∂v1

∂t= i

q1

m1

(

E +1cv1 ∧ B

)

,

ih∂v2

∂t= i

q2

m2

(

E +1cv2 ∧ B

)

.

We introduce a column of unknowns Φ = (E,W,v1,v2)T with 12 components. The preceding systemthen becomes

ih∂Φ∂t

= AΦ, A = A(x,−ih∇, h) = A0(x,−ih∇) + O(h2),

where A(x, p) is the matrix-valued symbol of the operator A:

A(x, p)Φ =

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

−W

(−ih∇) ∧ (−ih∇) ∧ E− n1q21

m1

(

E +1cv1 ∧ B

)

− n2q22

m2

(

E +1cv2 ∧ B

)

iq1

m1

(

E +1cv1 ∧ B

)

iq2

m2

(

E +1cv2 ∧ B

)

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

(we include all terms in A0 assuming that A1 = 0).We now study the stationary solutions. For this, we set

Φ = e−iωt/hΨ(x) ≡ e−iωt/h(W(x),E(x),v1(x),v2(x))T.

Omitting the primes and setting p = −ih∇ and ω2j = njq

2j /mj, we obtain

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

−W

p ∧ p ∧ E− ω21

(

E − 1cB ∧ v1

)

− ω22

(

E − 1cB ∧ v2

)

iq1

m1

(

E− 1cB ∧ v1

)

iq2

m2

(

E− 1cB ∧ v2

)

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

= ω

⎛

⎜

⎜

⎜

⎜

⎝

E

W

v1

v2

⎞

⎟

⎟

⎟

⎟

⎠

. (23)

5.2. Reduction to a pencil with a 3×3 symbol. We now eliminate W and vj . We have

W = −ωE, B ∧ vj

c− i

σj

vj

c= E, (24)

whereσj =

qj

mjωc. (25)

We make the following assumption.

1774

Assumption 4. We have σ2jB

2 �= 1 for all x.

Equation (24) then implies

vj

c=

11 − σ2

j B2

(

i(1 − σ2j B

2)σjE + σ21B ∧ E− iσ3

jB ∧ B ∧E)

. (26)

Eliminating vj and W from (23), we obtain the (already three-dimensional) system

p ∧ p ∧ E + ω2E + iv1

cω2

1

σ1+ i

v2

cω2

2

σ2= 0,

where v1 and v2 are determined by (26).We finally obtain the pencil of operators

H(x, p, ω)E ≡ p ∧ p ∧ E + aE + ibB ∧ E + cB ∧ B ∧ E = 0, (27)

where

a = ω2 − ω2pl, b =

ω21σ1

1 − σ21B2

+ω2

2σ2

1 − σ22B2

, c =ω2

1σ21

1 − σ21B2

+ω2

2σ22

1 − σ22B2

, (28)

and ωpl =√

ω21 + ω2

2 is the plasma frequency. It is easy to see that the operator H is at least symmetric.

5.3. Some properties of the symbol H(x, p, ω). We now study the properties of the matrix

H(x, p) = p ∧ p ∧ +aI + ibB ∧ +cB ∧ B∧ = 0.

Assumption 5. In the domain that interests us in the phase space, the vectors p and B are not

parallel.

The triple of p, B, and p ∧B then forms a basis, and E can hence be expanded as

E = z1p + z2B + z3p ∧ B. (29)

Proposition 1. Let Assumptions 4 and 5 be satisfied. The determinant H(x, p, ω) = detH and the

sum of the second-order principal minors I(x, p, ω) then become

H(x, p, ω) = a3 − 2a2cB2 − 2a2p2 − ab2B2 + ac2B4 + 3acp2B2 − ac〈p,B〉2 +

+ ap4 + b2p2B2 − b2〈p,B〉2 − c2p2B4 +

+ c2〈p,B〉2B2 − cp4B2 + cp2〈p,B〉2,

I(x, p, ω) = − a2 + 2a(p2 + cB2) + b2B2 − c2B4 − cp2B2 − c〈p,B〉2 − p4,

(30)

where a = a(ω), b = b(ω), and c = c(ω) can be obtained from formulas (28) and (25).

Proof. It is easy to see that the identities

p ∧ p ∧ p = 0, p ∧ p ∧ B = 〈p,B〉p − p2B, p ∧ p ∧ p ∧ B = −p2p ∧ B,

B ∧ p = −p ∧ B, B ∧ B = 0, B ∧ p ∧ B = B2p − 〈p,B〉B,

B ∧ B ∧ p = 〈p,B〉B − B2p, B ∧ B ∧ B = 0, B ∧B ∧ p ∧ B = −B2p ∧ B

1775

hold. Substituting (29) in (27) and using these identities, we obtain

z2(〈p,B〉p − p2B) − z3p2p ∧B + az1p + az2B + az3p ∧ B−

− ibz1p ∧ B + ibz3(B2p − 〈p,B〉B) + cz1(〈p,B〉B − B2p) − cz3B2p ∧ B =

= (z2〈p,B〉 + az1 + ibz3B2 − cz1B2)p + (−z2p2 + az2 + cz1〈p,B〉 − ibz3〈p,B〉)B +

+ (−z3p2 + az3 − ibz1 − cz3B2)p ∧ B.

Therefore,z1(a − cB2) + z2〈p,B〉 + ibz3B2 = 0,

z1c〈p,B〉 + z2(a − p2) − z3ib〈p,B〉 = 0,

ibz1 + z3(p2 − a + cB2) = 0.

(31)

The determinant of this system has form (30). The statement is proved.

If we now assume that H = 0 and I �= 0 (which is equivalent to Assumption 1), then we easily obtainthe eigenvector χr corresponding to the mode Hr. This vector is a nontrivial solution of the equationHχr = 0 (this solution is unique up to multiplication by a scalar). We solve (31) to obtain this solution.We can assume that a − p2 − cB2 �= 0 without loss of generality and can then express z2 and z3 in termsof z1 as

z2 = −(

b2

a − p2 − cB2+ c

)

〈p,B〉a − p2

z1, z3 =ib

a − p2 − cB2z1,

χr ≡ E = z1

(

p −(

b2

a − p2 − cB2+ c

)

〈p,B〉a − p2

B +ib

a − p2 − cB2p ∧ B

)

.

The normalization condition implies that

z1 =(

p2 +(

c +b2

a − p2 − cB2

)2 B2〈p,B〉2(a − p2)2

+b2

(a − p2 − cB2)2(p ∧ B)2

)−1/2

.

We have thus obtained explicit formulas for the determinant H and the eigenvectors χr.

5.4. Explicit form of the dispersion relation with regard to the physical values of theparameters. We use a specific example to show that the asymptotic form of stationary solutions can becalculated effectively. We consider a magnetic field of the form (in toroidal coordinates (15))

B =

⎛

⎜

⎜

⎝

Br(r, θ)

Bθ(r, θ)

0

⎞

⎟

⎟

⎠

≡ B0

a2 + r2 − 2ar cos θ

⎛

⎜

⎜

⎝

−a sin θ

r − a cos θ

0

⎞

⎟

⎟

⎠

.

Then

P2 = p2r +

p2θ

r2+

p2φ

Q2,

B2 = B2r (r, θ) + B2

θ (r, φ) =B2

0

a2 + r2 − 2ar cos θ,

〈P,B〉 =B0

a2 + r2 − 2ar cos θ

(

−apr sin θ +(r − a cos θ)pθ

r

)

.

1776

We note that pφ is a first integral.We take the values of parameters and constants typical of a real tokamak (see [5]): B0 = 3.4 · 104 G,

30 cm = a ≤ R0 = 90 cm, the electron charge qe = −4.8 · 10−10 CGSE, the proton charge qp = 4.8 ·10−10 CGSE, the electron mass me = 9.1 · 10−28 g, the proton mass mp = 1.672 · 10−24 g, the speedof light c = 2.99790 · 1010 cm/s, and the electron and proton densities n(r) = 1014N(r) and N(r) =N0(1 − r2/a2) + N1, where N0 = 1 and N1 = 0.5. Therefore, n(r) = 1.5 − r2/a2 = 1.5 − r2/302. Weintroduce the notation

Ω1 = Ωe =qe

mec= − q

mec, Ω2 = Ωp =

qp

mpc=

q

mpc, q = 4.8 · 10−10

and

α1 = ω21Ω

21 + ω2

2Ω22, α2 = ω2

2Ω21 + ω2

1Ω22,

β = (ω22Ω1 + ω2

1Ω2)2, γ = ω21ω

22(Ω1 − Ω2)2.

Hamiltonian (30) can then be rewritten as

H(p, x, ω) = ω2P2B2〈P,B〉2(ω2pl)Ω

21Ω

22 −

− ω4[〈P,B〉2P2α1 + B2(ω2pl)β + B2〈P,B〉2(ω2

pl)Ω21Ω

22 + B2P2(2β + γ) +

+ B2P4α2 + B4P2(ω2pl)Ω

21Ω

22 + B4P4Ω2

1Ω22 + 〈P,B〉2γ] +

+ ω6[(P4(ω2pl) + (ω2

pl)3 + 2P2(ω2

pl)2 + B2P4Ω1

2 + B2P2(α1 + 4α2) +

+ 〈P,B〉2α1 + B2(3β + 2γ) + 2B4P2Ω21Ω

22 + B4(ω2

pl)Ω21Ω

22] −

− ω8[P4 + 4P2(ω2pl) + 3(ω2

pl)2 + 2B2P2Ω1

2 + B2(α1 + 3α2) + B4Ω21Ω

22 +

+ ω10[2P2 + 3(ω2pl) + B2Ω2

1] − ω12 = 0. (32)

To use the results obtained in Sec. 4.3, we determine the minimums of Hamiltonian (32). We set pr = pθ = 0and θ = 0 (the case θ = π can be considered similarly). We then have

∂H∂pr

=∂H∂pθ

=∂H∂θ

= 0.

To determine the minimums, we must obtain the critical points of the function of r (we ignore the dependenceon ω and pφ):

H0(r) = H∣

∣

∣

∣

pr=pθ=θ=0

= − ω4

[

βω2plB

20

(a − r)2+

(2β + γ)p2φB2

0

(a − r)2(R0 + r)2+

+α2p

4φB2

0

(R0 + r)4(a − r)2+

ω2plΩ

21Ω

22B

40p2

φ

(a − r)4(R0 + r)2+

Ω21Ω

22p

4φB4

0

(R0 + r)4(a − r)4

]

+

+ ω6

[

ω2plp

4φ

(R0 + r)4+ ω6

pl +2ω4

plp2φ

(R0 + r)2+

Ω21p

4φB2

0

(R0 + r)4(a − r)2+

(α1 + 4α2)p2φB2

0

(a − r)2(R0 + r)2+

1777

+(3β + 2γ)B2

0

(a − r)2+

2Ω21Ω2

2B40p2

φ

(a − r)4(R0 + r)2+

ω2plΩ

21Ω2

2B40

(a − r)4

]

−

− ω8

[

p4φ

(R0 + r)4+

4ω2plp

2φ

(R0 + r)2+ 3ω4

pl +

+2Ω2

1p2φB2

0

(a − r)2(R0 + r)2+

(α1 + 3α2)B20

(a − r)2+

Ω21Ω2

2B40

(a − r)4

]

+

+ ω10

[ 2p2φ

(R0 + r)2+ 3ω2

pl +Ω2

1B20

(a − r)2

]

− ω12.

ThenH = H0(r) + Apr (r)p2

r + Apθ(r)p2

θ + Aθ(r)θ2 + O(

(p2r + p2

θ + θ2)2)

,

where

Apr = − ω4

[

(2β + γ)B20

(a − r)2+

2α2B20p2

φ

(R0 + r)2(a − r)2+

ω2plΩ

21Ω2

2B40

(a − r)4+

2Ω21Ω2

2B40p2

φ

(R0 + r)2(a − r)4

]

+

+ ω6

[ 2ω2plp

2φ

(R0 + r)2+ 2ω4

pl +2Ω2

1B20p2

φ

(R0 + r)2(a − r)2+

(α1 + 4α2)B20

(a − r)2+

2Ω21Ω

22B

40

(a − r)4

]

−

− ω8

[

2φ2

(R0 + r)2+ 4ω2

pl +2Ω2

1B20

(a − r)2

]

+ 2ω10,

Apθ= ω2

ω2plΩ

21Ω

22B

40p2

φ

r2(a − r)4(R0 + r)2− ω4

[

α1B20p2

φ

r2(R0 + r)2(a − r)2+

ω2plΩ

21Ω

22B

40

r2(a − r)4+

+(2β + γ)B2

0

r2(a − r)2+

2α2B20p2

φ

r2(R0 + r)2(a − r)2+

ω2plΩ

21Ω

22B

40

r2(a − r)2+

2Ω21Ω

22B

40p2

φ

r2(R0 + r)2(a − r)4+

+γB2

0

r2(a − r)2

]

+ ω6

[ 2ω2plp

2φ

r2(R0 + r)2+

2ω4pl

r2+

2Ω21B

20p2

φ

r2(R0 + r)2(a − r)2+

+(α1 + 4α2)B2

0

r2(a − r)4+

α1B20

r2(a − r)2+

2Ω21Ω

22B

40

r2(a − r)4

]

−

− ω8

[ 2p2φ

r2(R0 + r)2+

4ω2pl

r2+

2Ω21B

20

r2(a − r)2

]

+ ω10 2r2

,

Aθ = − ω4

[

−aβω2

plrB20

(a − r)4−

(2β + γ)r(a2 − 3ar + r2 − aR0)B20p2

φ

(a − r)4(R0 + r)3+

+α2r(2a2 − 5ar + 2r2 − aR0)B2

0p4φ

(R0 + r)5(a − r)4+

ω2plΩ

21Ω

22r(a

2 − 4ar + r2 − 2aR0)B40p2

φ

(a − r)6(R0 + r)3+

+2Ω2

1Ω22r(a

2 − 3ar + r2 − aR0)B40p4

φ

(R0 + r)5(a − r)6

]

+

+ ω6

[ 2ω2plrp

4φ

(R0 + r)5+

2ω4plrp

2φ

(R0 + r)3+

Ω21r(a

2 − 4ar + r2 − 2aR0)B20p4

φ

(R0 + r)5(a − r)4+

+(α1 + 4α2)r(a2 − 3ar + r2 − aR0)B2

0p2φ

(a − r)4(R0 + r)3− (3β + 2γ)arB2

0

(a − r)4+

1778

Fig. 1. Plot of H0(r) for pφ = 1011 and ω = 5.7 · 1011 with the scale on the vertical axis [−10140 , 10140].

Fig. 2. Plot of H0(r) for pφ = 1011 and ω = 5.67 · 1011 with the scale on the vertical axis [−10139 , 10139].

+2Ω2

1Ω22r(a

2 − 4ar + r2 − 2aR − 0)B40p2

φ

(a − r)6(R0 + r)3−

2ω2plΩ

21Ω

22arB4

0

(a − r)6

]

−

− ω8

[ 2rp4φ

(R0 + r)5+

4ω2plrp

2φ

(R0 + r)3+

2Ω21r(a

2 − 3ar + r2 − aR0)B20p2

φ

(a − r)4(R0 + r)3−

− (α1 + 3α2)arB20

(a − r)4− 2Ω2

1Ω22arB4

0

(a − r)6

]

+ ω10

[ 2rp2φ

(R0 + r)3− Ω2

1arB20

(a − r)4

]

− ω12.

To determine the values of the parameters corresponding to the minimums of H, it now suffices toanalyze functions of a single variable. Our numerical calculations show that such values of pφ and ω canbe determined. Figures 1–3 show that for φ = 1011, there exists a value ω ∈ [5.67 · 1011, 5.7 · 1011] such thatthe local minimum of H0(r) is attained for H0 = 0. It is also easy to verify that the functions Apr , Apθ

,and Aθ are positive and the minimum is hence nondegenerate.

Theorem 4 (together with formula (20)) can thus be used to calculate the asymptotic form of stationarysolutions. The choice of the parameters for satisfying the quantization condition and the explicit calculationof the spectral series and asymptotic solutions are interesting and nontrivial problems that we will considerin our further studies.

1779

Fig. 3. Plot of H0(r) for pφ = 1011 and ω = 5.7 · 1011 with the scale on the vertical axis [−10139 , 10139].

Appendix A: Proof of Theorem 1

We write the transport equation on a Lagrangian manifold Λ ⊂ {Hr = 0} as

dAλ

dt− 1

2tr

∂2Hr

∂p ∂xAλ + iLrAλ − λ

∂Hr

∂EAλ = 0,

where d/dt = adHr and

Lr = 〈χr,H1χr〉 − i

⟨

χr,dχr

dt

⟩

− i˜Lr, ˜Lr =⟨

χr,

n∑

j=1

biggl(∂H0

∂pj− ∂Hr

∂pj

)

∂χr

∂xj

⟩

.

Lemma 4. We have the equality

2Re˜Lr +⟨

χr,

n∑

j=1

(

∂2H0

∂pj ∂xj− ∂2Hr

∂pj ∂xj

)

χr

⟩

= 0.

Proof. We differentiate the identity 〈χr, (H0 − Hr)χr〉 = 0 with respect to pj and use the symmetryof the matrix H0:

⟨

χr,

(

∂H0

∂pj− ∂Hr

∂pj

)

χr

⟩

= 0. (33)

We differentiate the obtained relation with respect to xj and then sum the results over j to obtain therequired statement. The lemma is proved.

We apply this lemma to obtain

Lr = −i

⟨

χr,dχr

dt

⟩

+ 〈χr,H1χr〉 +i

2

⟨

χr,

n∑

j=1

∂2H0

∂pj ∂xjχr

⟩

− i

2tr

∂2Hr

∂p ∂x+ Im˜Lr.

We finally express Hr in terms of H = detH0, i.e., Hr = FrH, and take the Hamiltonian time along H ass to rewrite the transport equation as

FdAλ

ds+ F

⟨

χr,dχr

ds

⟩

Aλ + i〈χr,H1χr〉Aλ − 12

⟨

χr,

n∑

j=1

∂2H0

∂pj ∂xjχr

⟩

Aλ +

+ i Im⟨

χr,n∑

j=1

(

∂H0

∂pj− Fr

∂Hr

∂pj

)

∂χr

∂xj

⟩

Aλ − Fλ∂H∂E

Aλ = 0,

as required.

1780

Appendix B: Proof of Lemma 1

It suffices to verify that the function ξ constructed by formula (13) is a solution of system (10). It isclear that it is a Floquet solution by construction.

We note that

dvH(ZH(t))dt

= ΩHvH(ZH(t)),dvH(ZH(s))

ds= ΩHvH(ZH(s)).

For any function ξ such that dH(ξ) = 0, we further have

ΩH(t)ξ = FΩH∣

∣

z=ZH(t)ξ + Ω1ξ, Ω1 =

(

−Hx ⊗ Fp −Hx ⊗ Fx

Hp ⊗ Fp Hp ⊗ Fx

)

∣

∣

∣

∣

z=ZH (t)

.

In invariant form, Ω1ξ = dF (ξ)vH.Substituting (13) in (10), we obtain

dξ

dt− ΩHξ = F

dη

ds+ F

dr

dsvH + Fr

dvHds

− (FΩH + Ω1)(η + rvH) =

= F

(

dη

ds− ΩHη

)

+ Fr

(

dvHds

− ΩHvH

)

+(

Fdr

ds− dF (η + rvH)

)

vH =

= F

(

dr

ds− r

{F,H}F

− dF (η)F

)

vH.

It follows from condition (12) that this expression is zero. The lemma is proved.

Acknowledgments. The authors thank V. E. Nazaikinskii and I. A. Bogaevskii for the discussionsand useful remarks.

REFERENCES

1. V. P. Maslov and M. V. Fedoryuk, Semiclassical Approximation in Equations of Quantum Mechanics [in Russian],

Nauka, Moscow (1976); English transl.: Semiclassical Approximation in Quantum Mechanics (Math. Phys.

Appl. Math., Vol. 7), D. Reidel, Dordrecht (1981).

2. S. Yu. Dobrokhotov, D. S. Minenkov, and M. Rouleux, “The Maupertuis–Jacobi principle for Hamiltonians

of the form F (x, |p|) in two-dimensional stationary semiclassical problems,” Math. Notes, 97, 42–49 (2015);

A. Yu. Anikin, J. Bruning, and S. Yu. Dobrokhotov, “Averaging and trajectories of a Hamiltonian system

appearing in graphene placed in a strong magnetic field and a periodic potential,” Fundam. Prikl. Mat., 20,

5–20 (2015).

3. B. Yu. Sternin and V. E. Shatalov, “Characteristic Cauchy problem on a complex-analytic manifold,” in: Global

Analysis Studies and Applications I (Lect. Notes. Math., Vol. 1108, Yu. G. Borisovich, Yu. E. Gliklikh, and

A. M. Vershik, eds.), Springer, Berlin (1984), pp. 54–76.

4. I. A. Bogaevsky, “Caustics of interior scattering,” Proc. Steklov Inst. Math., 267, 1–7 (2009).

5. R. A. Gerwin, “Initial value solution of Maxwell’s equations in cold plasma,” Amer. J. Phys., 30, 711–715 (1962);

E. Mazzucato, “Propagation of a Gaussian beam in a nonhomogeneous plasma,” Phys. Fluid B: Plasma Phys.,

1, 1855–1859 (1989); Erratum, 2, 228 (1990); J. P. Freidberg, Ideal MHD, Cambridge Univ. Press, Cambridge

(2014); G. V. Pereverzev, “Paraxial WKB description of short wavelength eigenmodes in a tokamak,” Phys.

Plasmas, 8, 3664–3672 (2001); Yu. A. Kravtsov and P. B. Berczynski, “Gaussian beams in inhomogeneous

media: A review,” Stud. Geophys. Geod., 51, 1–36 (2007); R. A. Cairns and V. Fuchs, “Calculation of a wave

field from ray tracing,” Nucl. Fusion, 50, 095001 (2010); S. Yu. Dobrokhotov, A. Cardinali, A. I. Klevin, and

B. Tirozzi, “Maslov complex germ and high-frequency Gaussian beams for cold plasma in a toroidal domain,”

Dokl. Math., 94, 480–485 (2016).

1781

6. A. A. Shkalikov, “Operator pencils arising in elasticity and hydrodynamics: The instability index formula,”

in: Recent Developments in Operator Theory and Its Applications (Operator Theory: Adv. Appl., Vol. 87,

I. Gohberg, P. Lancaster, and P. N. Shivakumar, eds.), Birkhauser, Basel (1996), pp. 358–385.

7. V. P. Maslov, The Complex WKB Method for Nonlinear Equations [in Russian], Nauka, Moscow (1977); En-

glish transl. (Progr. Math. Phys., Vol. 16), Birkhauser, Basel (1994); V. V. Belov and S. Yu. Dobrokhotov,

“Semiclassical Maslov asymptotics with complex phases: I. General approach,” Theor. Math. Phys., 92, 843–

868 (1992); V. V. Belov, O. S. Dobrokhotov, and S. Yu. Dobrokhotov, “Isotropic tori, complex germ and Maslov

index, normal forms and quasimodes of multidimensional spectral problems,” Math. Notes, 69, 447–466 (2001);

S. Yu. Dobrokhotov and A. I. Shafarevich, “Semiclassical quantization of isotropic manifolds of Hamiltonian

systems,” in: Topological Methods in Theory of Hamiltonian Systems [in Russian], Faktorial, Moscow (1998),

pp. 41–114.

8. V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Short-Wave Diffraction Problems [in Russian], Nauka,

Moscow (1972); J. V. Ralston, “On the construction of quasimodes associated with stable periodic orbits,”

Commun. Math. Phys., 51, 219–242 (1976).

9. V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973); English transl.: Operational Methods,

Mir, Moscow (1976).

10. V. V. Belov, S. Yu. Dobrokhotov, and T. Ya. Tudorovskiy, “Operator separation of variables for adiabatic

problems in quantum and wave mechanics,” J. Eng. Math., 55, 183–237 (2016); S. B. Leble, Nonlinear Waves

in Waveguides with Stratification, Springer, Berlin (1991).

11. V. I. Arnol’d, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1974); English transl.

(Grad. Texts Math., Vol. 60), Springer, New York (1989); R. Abraham and J. E. Marsden, Foundations of Me-

chanics, Benjamin/Cummings, Reading, Mass. (1978); A. V. Tsiganov, “The Maupertuis principle and canonical

transformations of the extended phase space,” J. Nonlinear Math. Phys., 8, 157–182 (2001); S. Yu. Dobrokhotov

and M. Rouleux, “The semi-classical Maupertuis–Jacobi correspondence for quasi-periodic Hamiltonian flows

with applications to linear water waves theory,” Asymptotic. Anal., 74, 33–73 (2011); S. Yu. Dobrokhotov

and M. Rouleux, “The semiclassical Maupertuis–Jacobi correspondence and applications to linear water waves

theory,” Math. Notes, 87, 430–435 (2010).

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