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The Complex WKB Method for Nonlinear Equations I : Linear Theory - V. P. Maslov

The Complex WKB Method for Nonlinear Equations I

Linear Theory

By: V. P. Maslov, M.A. Shishkova (Translator), A. B. Sossinsky (Translator)

Hardcover

Published: 1st August 1994
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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 functions cp(x) exp{iS(x)/h}, x E 1R3, where S(x) is complex, and ImS(x) ~ o. 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.

Part 1 Equations and problems of narrow beam mechanics: asymptotic solutions of narrow beam type for partial differential equations with small parameter; systems of canonical equations; inequalities of the garding type; approximate solutions of the canonical system; generalized Cauchy problem and nonstationary transport equation. Part 2 Hamiltonian formalism of narrow beams: model problem; auxiliary facts from symplectic geometry of the phase space; Lagrangian manifolds with real germ; phase and action on Lagrangian manifolds with real germ; phase reconstruction; Lagrangian manifolds with complex germ; dissipation conditions; action on Lagrangian manifolds with complex germ; canonical transformations of Lagrangian manifolds with complex germ; approximate complex solutions of the nonstationary Hamilton-Jacobi equation. Part 3 Approximate solutions of the nonstationary transport equation: approximate real solutions of the transport equation; approximate complex solutions of the nonstationary transport equation; creation and annihilation operators for the generalized nonstationary transport equation; creation and annihilation operators - general case; the spaces of functions S([lambdak, gamman/T lambdak]); generalized transport equation with nonzero right-hand side. Part 4 Stationary Hamilton-Jacobi and transport equations: canonical system of stationary equations; invariant Lagrangian manifolds with complex germ; approximate solutions of the stationary Hamilton-Jacobi equation and the transport equation; the generalized Cauchy problem for stationary Hamilton-Jacobi equations; the Cauchy problem in the plane for transport equations; generalized stationary transport equation; examples; generalized eigenfunctions of the Helmholtz operator. Part 5 Complex Hamiltonian formalism of compact (cyclic) beams: setting the problem; invariant zero-dimensional Lagrangian manifolds with complex germ; approximate solutions of the generalized transport equation concentrated in the neighborhood of a point; family of closed curves with complex germ; functions on a family of closed curves with complex germ; creation operators; invariant closed curves with complex germ; approximate cyclic solutions of the stationary Hamilton-Jacobi equation; approximate solutions of the generalized transport equation. Part 6 Canonical operators on Lagrangian manifolds with complex germ and their applications to spectral problems of quantum mechanics: invariant closed curves with complex germ in systems with one cyclic variable; semiclassical spectral series for Schrodinger and Klein-Gordon operators in electromagnetic fields with axial symmetry corresponding to relative equilibrium positions; construction of the canonical operator on Lagrangian manifolds with complex germ; canonical operators and polynomial beams over isotropic manifolds; example; table of asymptotic spectral series.

ISBN: 9783764350888
ISBN-10: 3764350881
Series: Progress in Mathematical Physics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 304
Published: 1st August 1994
Publisher: Birkhauser Verlag AG
Country of Publication: CH
Dimensions (cm): 23.5 x 15.5  x 1.9
Weight (kg): 1.37