Kam Theory and Semiclassical Approximations to Eigenfunctions

ISBN-13: 9783642762499 / Angielski / Miękka / 2011 / 387 str.

Vladimir F. Lazutkin; A. I. Shnirelman

Opis

It is a surprising fact that so far almost no books have been published on KAM theory. The first part of this book seems to be the first monographic exposition of this subject, despite the fact that the discussion of KAM theory started as early as 1954 (Kolmogorov) and was developed later in 1962 by Arnold and Moser. Today, this mathematical field is very popular and well known among physicists and mathematicians. In the first part of this Ergebnisse-Bericht, Lazutkin succeeds in giving a complete and self-contained exposition of the subject, including a part on Hamiltonian dynamics. The main results concern the existence and persistence of KAM theory, their smooth dependence on the frequency, and the estimate of the measure of the set filled by KAM theory. The second part is devoted to the construction of the semiclassical asymptotics to the eigenfunctions of the generalized Schrodinger operator. The main result is the asymptotic formulae for eigenfunctions and eigenvalues, using Maslovs operator, for the set of eigenvalues of positive density in the set of all eigenvalues. An addendum by Prof. A.I. Shnirelman treats eigenfunctions corresponding to the "chaotic component" of the phase space."

Spis treści

List of General Mathematical Notations.- I. KAM Theory.- I. Symplectic Dynamical Systems.- §1. Symplectic Vector Spaces.- §2. Symplectic Manifolds.- §3. Symplectic Dynamical Systems.- §4. Symplectic Gluing.- §5. Cross-sections.- §6. Generalized Geodesic Flows.- §7. Completely Integrable Hamiltonian Systems.- §8. Systems in an Annulus.- Notes to Chapter I.- II. KAM Theorems.- §9. The KAM Torus.- §10. KAM Set.- §11. The KAM Theorem in an Annulus.- §12. Near a Torus.- §13. Near a Periodic Motion.- §14. Near the Boundary of Planar Convex Billiards.- §15. The Robustness of a KAM Set.- Notes to Chapter II.- III. Beyond the Tori.- §16. General Picture of Stochasticity Near KAM Tori. The Case of More than Two Degrees of Freedom.- §17. Picture of Stochasticity Near KAM Tori in the Case of Two Degrees of Freedom.- Notes to Chapter III.- IV. Proof of the Main Theorem.- §18. Two Reductions.- §19. Machinery.- §20. Description of the Iterative Process.- §21. Reproduction of (20.1i and (20.2i. Convergence of Fi.- §22. Estimates of ?i+1.- §23. Reproduction of (20.3i).- §24. Reproduction of (20.4i).- §25. Convergence of the Process and the Estimate of ? ? — id ?.- §26. Derivatives of G at points of ?n × ?.- §27. The End of the Proof of Theorem 18.10.- §28. Deduction of the Theorem for Discrete Time from That of Continuous Time.- Notes to Chapter IV.- II. Eigenfunctions Asymptotics.- V. Laplace-Beltrami-Schrödinger Operator and Quasimodes.- §29. Basic Facts about Self-Adjoint Operators and Spectra.- §30. Laplace-Beltrami-Schrödinger Operator.- §31. Particular Cases.- §32. Quasimodes.- §33. Degenerated Quasimodes.- Notes to Chapter V.- VI. Maslov’s Canonical Operator.- §34. Assumptions.- §35. The Local Canonical Operator.- §36. The Commutation Rule.- §37. Theory of Maslov’s Indices.- §38. A Global Formula for Maslov’s Operator.- Notes to Chapter VI.- VII. Quasimodes Attached to a KAM Set.- §39. The Canonical Maslov’s Operator Associated with a KAM Set.- §40. Quantum Conditions and the Set ?.- §41. Construction of Quasimodes.- §42. Orthogonality.- Notes to Chapter VII.- Addendum (by A.I. Shnirelman). On the Asymptotic Properties of Eigenfunctions in the Regions of Chaotic Motion.- Appendix I. Manifolds.- Appendix II. Derivatives of Superposition.- Appendix III. The Stationary Phase Method.- References.