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KAM Theory and Semiclassical Approximations to Eigenfunctions

  • Vladimir F. Lazutkin

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 24)

Table of contents

  1. Front Matter
    Pages I-IX
  2. Introduction

    1. Vladimir F. Lazutkin
      Pages 1-7
  3. List of General Mathematical Notations

    1. Vladimir F. Lazutkin
      Pages 8-11
  4. KAM Theory

    1. Front Matter
      Pages 13-13
    2. Vladimir F. Lazutkin
      Pages 15-120
    3. Vladimir F. Lazutkin
      Pages 121-159
    4. Vladimir F. Lazutkin
      Pages 160-187
    5. Vladimir F. Lazutkin
      Pages 188-221
  5. Eigenfunctions Asymptotics

    1. Front Matter
      Pages 223-223
    2. Vladimir F. Lazutkin
      Pages 225-241
    3. Vladimir F. Lazutkin
      Pages 242-293
    4. Vladimir F. Lazutkin
      Pages 294-312
  6. Back Matter
    Pages 338-387

About this book

Introduction

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.

Keywords

Eigenvalue Hamilton-Systeme Hamiltonian systems KAM Theorie KAM theory convergence deduction derivative dynamical systems hamiltonian system manifold measure operator schrödinger equation schrödinger operator

Authors and affiliations

  • Vladimir F. Lazutkin
    • 1
  1. 1.Department of Mathematical Physics Institute of PhysicsSt. Petersburg State UniversityPetrodvorets St. PetersburgRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-76247-5
  • Copyright Information Springer-Verlag Berlin Heidelberg 1993
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-76249-9
  • Online ISBN 978-3-642-76247-5
  • Series Print ISSN 0071-1136
  • Buy this book on publisher's site