The Complex Wkb Method for Nonlinear Equations I: Linear Theory

ISBN-13: 9783034896696 / Angielski / Miękka / 2012 / 304 str.

Victor P. Maslov; A. B. Sossinsky; Victor P. Maslov

Opis

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.

Spis treści

I. Equations and problems of narrow beam mechanics.- II. Hamiltonian formalism of narrow beams.- III. Approximate solutions of the nonstationary transport equation.- IV. Stationary Hamilton-Jacobi and transport equations.- V. Complex Hamiltonian formalism of compact (cyclic) beams.- VI. Canonical operators on Lagrangian manifolds with complex germ and their applications to spectral problems of quantum mechanics.- References.- Appendix A Complex germ generated by a linear connection.- Appendix B Asymptotic solutions with pure imaginary phase and the tunnel equation.- Appendix C Analytic asymptotics of oscillatory decreasing type (heuristic considerations).