| The Complex Wkb Method for Nonlinear Equations I 1994 Edition Contributor(s): Maslov, Victor P. (Author), Shishkova, M. a. (Translator), Sossinsky, A. B. (Translator) |
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ISBN: 3764350881 ISBN-13: 9783764350888 Publisher: Birkhauser OUR PRICE: $52.24 Product Type: Hardcover - Other Formats Published: August 1994 Annotation: When this book was first published (in Russian), it proved to become the fountainhead of a major stream of important papers in mathematics, physics and even chemistry; indeed, it formed the basis of new methodology and opened new directions for research. The present English edition includes new examples of applications to physics, hitherto unpublished or available only in Russian. Its central mathematical idea is to use topological methods to analyze isotropic invariant manifolds in order to obtain asymptotic series of eigenvalues and eigenvectors for the multi-dimensional Schrodinger equation, and also to take into account the so-called tunnel effects. Finite-dimensional linear theory is reviewed in detail. Infinite-dimensional linear theory and its applications to quantum statistics and quantum field theory, as well as the nonlinear theory (involving instantons), will be considered in a second volume. |
| Additional Information |
| BISAC Categories: - Gardening - Science | Physics - Mathematical & Computational - Mathematics | Mathematical Analysis |
| Dewey: 515 |
| LCCN: 94022799 |
| Series: Molecular and Cell Biology Updates |
| Physical Information: 0.75" H x 6.14" W x 9.21" (1.36 lbs) 304 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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. |
