Spectral Asymptotics in the Semi-Classical Limit

This book presents the basic methods and applications in semiclassical approximation in the light of developments.

M. Dimassi (Author), J. Sjostrand (Author)

9780521665445, Cambridge University Press

Paperback, published 16 September 1999

240 pages
22.8 x 15.4 x 1.5 cm, 0.33 kg

'This book is an excellent introduction to a modern rapidly developing subject which lies between mathematics and physics.' Yuri Safarov, Bulletin of the London Mathematical Society

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.

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.

Subject Areas: Calculus & mathematical analysis [PBK]