Beschreibung
InhaltsangabeList 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.
