Spectral Asymptotics in the Semi-Classical Limit
Spectral Asymptotics in the Semi-Classical Limit
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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.
- Indispensable for researchers in this area
- Authors are top names
- Covers results from recent years
Reviews & endorsements
"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
Product details
April 2011Adobe eBook Reader
9780511893599
0 pages
0kg
This ISBN is for an eBook version which is distributed on our behalf by a third party.
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.
