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KAM Theory and Semiclassical Approximations to Eigenfunctions

  • Book
  • © 1993

Overview

Authors:
  1. Vladimir F. Lazutkin

    1. Department of Mathematical Physics Institute of Physics, St. Petersburg State University, Petrodvorets St. Petersburg, Russia

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Table of contents (10 chapters)

  1. Front Matter

    Pages I-IX
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  2. Introduction

    1. Introduction

      • Vladimir F. Lazutkin
      Pages 1-7

  3. List of General Mathematical Notations

    1. List of General Mathematical Notations

      • Vladimir F. Lazutkin
      Pages 8-11

  4. KAM Theory

    1. Front Matter

      Pages 13-13
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    2. Symplectic Dynamical Systems

      • Vladimir F. Lazutkin
      Pages 15-120

    3. KAM Theorems

      • Vladimir F. Lazutkin
      Pages 121-159

    4. Beyond The Tori

      • Vladimir F. Lazutkin
      Pages 160-187

    5. Proof of the Main Theorem

      • Vladimir F. Lazutkin
      Pages 188-221

  5. Eigenfunctions Asymptotics

    1. Front Matter

      Pages 223-223
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    2. Laplace-Beltrami-Schrödinger Operator and Quasimodes

      • Vladimir F. Lazutkin
      Pages 225-241

    3. Maslov’s Canonical Operator

      • Vladimir F. Lazutkin
      Pages 242-293

    4. Quasimodes Attached to a KAM Set

      • Vladimir F. Lazutkin
      Pages 294-312

    5. Addendum On the Asymptotic Properties of Eigenfunctions in the Regions of Chaotic Motion

      • A. I. Shnirelman
      Pages 313-337

  6. Back Matter

    Pages 338-387
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Keywords

About this book

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 Schrödinger operator. The main result is the asymptotic formulae for eigenfunctions and eigenvalues, using Maslov`s 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.

Authors and Affiliations

  • Department of Mathematical Physics Institute of Physics, St. Petersburg State University, Petrodvorets St. Petersburg, Russia

    Vladimir F. Lazutkin

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