Spectral Asymptotics in the Semi-Classical Limit

ISBN-13: 9780521665445 / Angielski / Miękka / 1999 / 240 str.

M. Dimassi; Mouez Dimassi; J. Sjostrand

Opis

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 Schrodinger 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.

Recenzje

'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

Spis treści

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.