The Complex WKB Method for Nonlinear Equations I

Linear Theory

Victor P. Maslov (Autor)

Buch | Hardcover

VII, 304 Seiten

1994 | 1994
Springer Basel (Verlag)
978-3-7643-5088-8 (ISBN)

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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.

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).

Erscheint lt. Verlag	1.8.1994
Reihe/Serie	Progress in Mathematical Physics
Übersetzer	M.A. Shishkova, A.B. Sossinsky
Zusatzinfo	VII, 304 p.
Verlagsort	Basel
Sprache	englisch
Maße	155 x 235 mm
Gewicht	621 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
	Naturwissenschaften ► Physik / Astronomie ► Allgemeines / Lexika
	Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
Schlagworte	eigenvalue • Eigenvector • Field Theory • instanton • Jacobi • manifold • Mathematica • Mechanics • quantum field • quantum field theory • quantum mechanics • Schrödinger equation • Statistics
ISBN-10	3-7643-5088-1 / 3764350881
ISBN-13	978-3-7643-5088-8 / 9783764350888
Zustand	Neuware