Spectral Asymptotics in the Semi-Classical Limit

by M. Dimassi, J. Sjostrand

Overview

Semiclassical approximation addresses the important relationship between quantum and classical mechanics. In recent years mathematical theory has undergone significant growth, mainly due to microlocal analysis techniques. This volume develops the basic methods of the theory, including the WKB-method, stationary phase and h-pseudodifferential operators. The authors employ the systematic use of a Cauchy formula that simplifies the functional calculus of pseudodifferential operators. The applications described include recent results on the tunnel effect, the asymptotics of eigenvalues in relation to classical trajectories and normal forms, plus slow perturbations of periodic Schrödinger operators appearing in solid state physics. The text assumes no previous specialized knowledge in quantum mechanics or microlocal analysis, and only general knowledge of spectral theory in Hilbert space, distributions, Fourier transforms and some differential geometry.

Product Details

ISBN-13:	9780521665445
Publisher:	Cambridge University Press
Publication date:	09/16/1999
Series:	London Mathematical Society Lecture Note Series , #268
Edition description:	New Edition
Pages:	240
Product dimensions:	6.06(w) x 8.98(h) x 0.59(d)

Table of Contents

Introduction; 1. Local symplectic geometry; 2. The WKB-method; 3. The WKB-method for a potential minimum; 4. Self-adjoint operators; 5. The method of stationary phase; 6. Tunnel effect and interaction matrix; 7. h-pseudodifferential operators; 8. Functional calculus for pseudodifferential operators; 9. Trace class operators and applications of the functional calculus; 10. More precise spectral asymptotics for non-critical Hamiltonians; 11. Improvement when the periodic trajectories form a set of measure 0; 12. A more general study of the trace; 13. Spectral theory for perturbed periodic problems; 14. Normal forms for some scalar pseudodifferential operators; 15. Spectrum of operators with periodic bicharacteristics; References; Index; Index of notation.

Editorial Reviews

"Researchers and graduate students in mathematical analysis are the intended readership, but anyone with an interest in the current state of the mathematics of quantum mechanics will get a lot out of this book." Proceedings of the Edinburgh Mathematical Society

"The authors have managed to write a very useful, clear, and readable book of quite moderate size, applying modern sophisticated methods in order to attack successfully some important, interesting, and difficult problems of contemporary mathematical physics." Mathematical Reviews

From the Publisher

A textbook developed from a one-semester course for graduates and researchers in mathematical analysis and physics. Dimassi (U. de Paris-Nord) and Sj<:o>strang (Ecole Polytechnique) include more recent material than classical studies of spectral asymptotics, but attempt to keep the treatment at an introductory level. Among their applications are results on the tunnel effect, the asymptotics of eigenvalues in relation to classical trajectories and normal forms, and the slow perturbation of periodic Schr<:o>dinger operators in solid-state physics. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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