Spectral asymptotics in the semi-classical limit / Mouez Dimassi, Johannes Sjöstrand.

Format:

Book

Author/Creator:

Dimassi, Mouez, author.

Sjöstrand, J. (Johannes), author.

Series:

London Mathematical Society lecture note series ; 268.

London Mathematical Society lecture note series ; 268

Language:

English

Subjects (All):

Microlocal analysis.

Quantum theory.

Approximation theory.

Spectral theory (Mathematics).

Mathematical physics.

Physical Description:

1 online resource (xi, 227 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 1999.

Language Note:

English

Summary:

Semiclassical approximation addresses the important relationship between quantum and classical mechanics. There has been a very strong development in the mathematical theory, mainly thanks to methods of microlocal analysis. This book develops the basic methods, including the WKB-method, stationary phase and h-pseudodifferential operators. The applications include 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. No previous specialized knowledge in quantum mechanics or microlocal analysis is assumed, and only general facts about spectral theory in Hilbert space, distributions, Fourier transforms and some differential geometry belong to the prerequisites. This book is addressed to researchers and graduate students in mathematical analysis, as well as physicists who are interested in rigorous results. A fairly large fraction can be (and has been) covered in a one semester course.

Contents:

Local symplectic geometry

The WKB-method

The WKB-method for a potential minimum

Self-adjoint operators

The method of stationary phase

Tunnel effect and interaction matrix

@h-pseudodifferential operators

Functional calculus for pseudodifferential operators

Trace class operators and applications of the functional calculus

More precise spectral asymptotics for non-critical Hamiltonians

Improvement when the periodic trajectories form a set of measure 0

A more general study of the trace

Spectral theory for perturbed periodic problems

Normal forms for some scalar pseudodifferential operators

Spectrum of operators with periodic bicharacteristics.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-139-88557-X

1-107-36770-0

1-107-37224-0

1-107-36279-2

1-107-36924-X

1-299-40530-4

1-107-36524-4

0-511-66219-X