Table Of ContentProgress in Physics
Volume 16
Victor P. Maslov
The Complex WKB Method
for Nonlinear Equations I
Linear Theory
Translated from the Russian by
M. A. Shishkova and
A. B. Sossinsky
Springer Buei AG
Author:
Victor P. Maslov
Department of Applied Mathematics
Moscow Institute of Electronics and Mathematics
B. Vuzovsky per., 3/12
Moscow 109028
Russia
Based on the book "Kompleksnyi metod VKB v nelineinyh uravnenijah", originally
published in Russian by Nauka.
A CIP catalogue record for this book is available from the Library of Congress,
Washington D. c., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Maslov, Viktor P.:
The complex WKB method for nonlinear equations / Viktor P.
Maslov. Transl. from the Russ. by M. A. Shishkova and
A. B. Sossinsky. - Basel ; Boston ; Berlin : Birkhauser.
Einheitssacht.: Kompleksnyj metod VKB v nelinejnych uravnenijach <dt.>
1. Linear theory. - 1994
(Progress in physics ; VoI. 16)
ISBN 978-3-0348-9669-6 ISBN 978-3-0348-8536-2 (eBook)
DOI 10.1007/978-3-0348-8536-2
NE:GT
This work is subject to copyright. AII rights are reserved, whether the whole or
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© 1994 Springer Basel AG
Originally published by Birkhăuser Verlag in 1994
Softcover reprint of the hardcover 1s t edition 1994
ISBN 978-3-0348-9669-6
987654321
CONTENTS
Introduction............................................ 1
Chapter I. Equations and problems of narrow beam
mechanics
§l. Asymptotic solutions of narrow beam type for partial
differential equations with small parameter ........ 7
§2. Systems of canonical equations .................... 12
§3. Inequalities of the Garding type ................... 16
§4. Approximate solutions of the canonical system ..... 18
§5. Generalized Cauchy problem and nonstationary
transport equation ................................ 25
Chapter II. Hamiltonian formalism of narrow beams
§l. Model problem. ................................... 31
§2. Auxiliary facts from symplectic geometry of the phase
space.............................................. 37
§3. Lagrangian manifolds with real germ .............. 39
§4. Phase and action on Lagrangian manifolds with real
germ.............................................. 46
§5. Phase reconstruction .............................. 51
§6. Lagrangian manifolds with complex germ .......... 55
§7. Dissipation conditions ............................. 58
§8. Action on Lagrangian manifolds with complex
germ.............................................. 60
§9. Canonical transformations of Lagrangian manifolds
with complex germ ................................ 62
§1O. Approximate complex solutions of the nonstationary
Hamilton-Jacobi equation ......................... 64
Chapter III. Approximate solutions of the
nonstationary transport equation
§l. Approximate real solutions of the transport
equation .......................................... 69
§2. Approximate complex solutions of the nonstationary
transport equation ................................ 73
vi Contents
§3. Creation and annihilation operators for the
generalized nonstationary transport equation 82
§4. Creation and annihilation operators. General case .. 91
§5. The spaces offunctions S([Ak,rnjTAk]) ........... 105
§6. Generalized transport equation with nonzero
right-hand side .................................... 107
Chapter IV. Stationary Hamilton-Jacobi and
transport equations
§l. Canonical system of stationary equations .......... 111
§2. Invariant Lagrangian manifolds with complex
germ.............................................. 113
§3. Approximate solutions of the stationary Hamilton-
Jacobi equation and the transport equation ........ 117
§4. The generalized Cauchy problem for stationary
Hamilton-Jacobi equations. . . . . . . . . . . . . . . . . . . . . . . . . 123
§5. The Cauchy problem in the plane for transport
equations ......................................... 134
§6. Generalized stationary transport equation ......... 136
§7. Examples ......................................... 141
§8. Generalized eigenfunctions of the Helmholtz
operator .................................. .'....... 144
Chapter V. Complex Hamiltonian formalism of
compact (cyclic) beams
§l. Setting the problem ............................... 155
§2. Invariant zero-dimensional Lagrangian manifolds
with complex germ ................................ 160
§3. Approximate solutions of the generalized transport
equation concentrated in the neighborhood
of a point ......................................... 166
§4. Family of closed curves with complex germ ........ 171
§5. Functions on a family of closed curves with complex
germ; creation operators ........................... 176
§6. Invariant closed curves with complex germ ......... 184
§7. Approximate cyclic solutions of the stationary
Hamilton-Jacobi equation ......................... 192
§8. Approximate solutions of the generalized transport
equation .......................................... 196
Chapter VI. Canonical operators on Lagrangian
manifolds with complex germ and their applica
tions to spectral problems of quantum mechanics
§l. Invariant closed curves with complex germ in systems
with one cyclic variable ............................ 203
Contents vii
§2. Semiclassical spectral series for Schrodinger and
Klein-Gordon operators in electromagnetic fields with
axial symmetry corresponding to relative equilibrium
posUions ............. .................... ......... 207
§3. Construction of the canonical operator on Lagrangian
manifolds with complex germ ...................... 212
§4. Canonical operators and polynomial beams over
isotropic manifolds ................................ 219
§5. Example .......................................... 227
§6. Table of asymptotic spectral series ................. 243
References .............................................. 251
Appendix A
Complex germ generated by a linear
connection ........................................... 257
Appendix B
Asymptotic solutions with pure imaginary phase
and the tunnel equation ............................ 267
Appendix C Analytic asymptotics of oscillatory
decreasing type (heuristic considerations) ........ 293
INTRODUCTION
This book deals with asymptotic solutions of linear and nonlinear equa
tions which decay as h ---+ 0 outside a neighborhood of certain points, curves
and surfaces. Such solutions are almost everywhere well approximated by the
o.
functions cp(x) exp{iS(x)/h}, x E 1R3, where S(x) is complex, and ImS(x) ~
When the phase S(x) is real (ImS(x) = 0), the method for obtaining asymp
totics of this type is known in quantum mechanics as the WKB-method. We
preserve this terminology in the case ImS(x) ~ 0 and develop the method for
a wide class of problems in mathematical physics.
Asymptotics of this type were constructed recently for many linear prob
lems of mathematical physics; certain specific formulas were obtained by differ
ent methods (V. M. Babich [5 -7], V. P. Lazutkin [76], A. A. Sokolov, 1. M. Ter
nov [113], J. Schwinger [107, 108], E. J. Heller [53], G. A. Hagedorn [50, 51],
V. N. Bayer, V. M. Katkov [21], N. A. Chernikov [35] and others). However,
a general (Hamiltonian) formalism for obtaining asymptotics of this type is
clearly required; this state of affairs is expressed both in recent mathematical
and physical literature. For example, the editors of the collected volume [106]
write in its preface: "One can hope that in the near future a computational pro
cedure for fields with complex phase, similar to the usual one for fields with real
phase, will be developed." It turns out that these asymptotics are well-defined
by approximate complex solutions of certain (ordinary) differential Hamilton
ian equations, namely, by the equations of the bicharacteristics associated with
the initial problem [87, 89]. In the present monograph the simplest version of
such a complex approximation theory (the theory of the "complex germ") is
presented in its simplest form for certain specific problems, especially spectral
problems.
The complex solutions of real analytic characteristics equations were con
sidered in the well-known book by Leray, Garding, Kotake [79] where the au
thors investigated singularities of analytic solutions for partial differential equa
tions. In the physical literature, complex solutions of real analytic Hamilton
Jacobi equations and Hamilton systems have been used for a long time (by
J. Keller in the diffraction problem [60], the author of this book in the scat
tering problem [85, 86] (also see [117]), Yu. Kravtsov in the similar diffraction
problem [68, 69], and others). This approach involves overcoming certain dif
ficulties arising when one tries to obtain analytic solutions of the Hamilton-
Typeset by AM5-1E;X
2 Introduction
Jacobi equation, as well as difficulties related to choosing the non-single-valued
solution branch correctly. These difficulties can be avoided by solving the prob
lem mod O(hN) and using constructions based on the following simple ideas.
The asymptotic solution is supposed to have the form exp{iS(x)/h}cp(x),
where ImS ~ 0 (the latter condition means that the solution is bounded
as h ---+ 0). Obviously, the values of the functions Sand cp in the domain
1m S ~ 8 > 0 are not essential, since the solution vanishes in this domain
within the accuracy considered. Hence the complex part of the formula S
is an additional small parameter. This can be also seen from the estimate
(ImSyr exp{ih-lS} = O(h'Y). Therefore, one can construct asymptotic analogs
of the analytic Hamiltonian formalism in which analyticity is replaced by al
most analyticity, i.e., where the Cauchy-Riemann conditions are required to
hold mod (1m syr .
This idea was used heuristically in the theory of complex germs, see [87],
and also provided a basis for the almost analytic formalism. The same idea
also appeared in [116J when the parametrix for equations of principal type was
constructed.
Since 1965, the main geometrical object in the theory of characteristics is
the n-dimensional submanifold of the phase space JR.~ x JR.; which annihilates
the symplectic form; it was called the Lagrangian manifold by the author.
The geometrical basis of the almost analytic formalism is the notion of
almost analytic Lagrangian manifold. Locally it is a 2n-dimensional real sub
c
manifold in the complex phase space 2n = JR.4n. This formalism is similar to
analytic theory and possesses a striking inner harmony [70, 87, 92, 97, 116].
The central geometrical construction of another approach is the Lagrangi
an manifold with complex germ. The complex germ structure is defined on an
n-dimensional real manifold A by the "almost Lagrangian" imbedding of it into
the complex phase space CQx C~. This means that a nonnegative function D
(the dissipation) and a function W satisfying the condition dW = P dQ+O(D)
are given on A. Here we assume that the dissipative condition is satisfied, which
means essentially that the planes tangent to A are C-Lagrangian on the zero-set
of dissipation, i.e., they are real-similar, annihilate the form dP I\dQ and satisfy
the following nonnegativity condition: Im(Po:g, Qo:g) ~ 0 for all 9 E cn, where
a = (al' ... ,an) are coordinates on the manifold A. In contrast to the almost
analytic theory, the formulas in complex germ theory are simpler and more
constructive (concerning the relationship with the almost analytic formalism,
see [36, 37]).
The general theory of complex germs was developed further in [25, 41,
70, 72, 96, 97, 121J and found its applications in mathematical problems con
cerning the stability of difference schemes in the Cauchy problem for complex
Hamiltonians [71, 96, 100, 114J and the solvability of differential and pseudo
differential equations [38, 42J. Nevertheless, this theory is rather burdensome
from the point of view of applications. The same can be said about the almost
Introduction 3
analytical formalism, whose main theorems are of conditional character. The
construction of the main object in this theory, namely, the invariant complex
Lagrangian manifold, is based on complex solutions of nonlinear differential
equations which have not been studied well enough until now. In particular, it
is not effective for investigating the spectra of operators.
On the other hand, in many specific important problems, the characteris
tics of the initial equations are real and yield a simpler construction. The crucial
point in it is the Hamiltonian approach to the classical mechanics of infinitely
narrow beams giving complex solutions of the Hamilton-Jacobi (eikonal) equa
tion and of the transport (Liouville) equation.
This construction is satisfactory not only because of the comparative sim
plicity of the final physical statements for certain problems in quantum physics,
as well as some in other branches of physics and mechanics. For example, such
problems have become essential in wave optics, i.e., in the problem of "narrow
beams", in connection with the intensive development of laser techniques.
The geometric objects of the phase space introduced in this book, namely,
the isotropic (incomplete Lagrangian) manifolds with complex germ, are the
mathematical equivalent of the physical notion of strongly localized (concen
trated) wave fields in a neighborhood of a beam which is precisely the one
dimensional isotropic manifold.
In quantum mechanics, such localized wave fields are associated with
the notion of Gaussian wave packet with center of gravity moving along the
classical trajectories of charged particles. Using the theory of the complex
germ, complete sets of such states (called classical trajectory-coherent) for the
Schr6dinger, Klein-Gordon, Pauli, Dirac and Dirac-Pauli operators in an arbi
trary electromagnetic field, for the Dirac operator in a torsion field and in an
arbitrary gauge Yang-Mills field with symmetry group SU(2) were constructed
[8, 10, 13, 18-20, 24, 29-31J. These states have the following characteristic
properties: the quantum mechanical average of the coordinates and momenta
operators satisfy the classical Newton and Lorentz equations respectively as
h --+ 0 [12, 14J.
For nonrelativistic quadratic quantum systems, the semiclassical trajecto
ry-coherent states are exact solutions of the corresponding dynamical equations
of motion and coincide in this case with the so-called coherent states [47, 62,
83, 102, 105]' which were already introduced in 1926 by E. Schr6dinger in quan
tum theory [109]. The complete sets of semiclassical trajectory-coherent states
constructed appear to be convenient for calculating certain specific effects of
quantum electrodynamics in the Farry representation [80]; in the corresponding
description of a charge's interaction with the electromagnetic field, the external
(classical) field is taken into account exactly, while the quantized radiation field
is dealt with according to the perturbation theory.
In particular, when the spontaneous radiation of bosons was calculated
by means of such states, a regular expansion of the radiation characteristics as
h --+ 0 was obtained for arbitrary external fields [9, 11, 17]. The zero term of
