The complex WKB method for nonlinear equations. Progress in physics; Bd. 16.

Maslov, Victor P.

ISBN 10: 3764350881 / ISBN 13: 9783764350888

Editore: Birkhäuser, 1994

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VII, 300 S. Ex-library with stamp and library-signature. GOOD condition, some traces of use. C-03762 3764350881 Sprache: Englisch Gewicht in Gramm: 550.

Codice articolo 2489688

Dati bibliografici

• Titolo: The complex WKB method for nonlinear equations. Progress in physics; Bd. 16.
• Editore: Birkhäuser
• Data di pubblicazione: 1994
• Lingua: Inglese
• ISBN 10: 3764350881
• ISBN 13: 9783764350888
• Rilegatura: Hardcover
• Cataloghi dei venditori:
  • SA MATHEMATIK

Descrizione articolo

Riassunto

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.

Contenuti

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).