KAM theory and semiclassical approximations to eigenfunctions. With an addendum by A. I. Shnirelman. (English) Zbl 0814.58001

Lazutkin, Vladimir F.

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 24. Berlin: Springer-Verlag. ix, 387 p. (1993).

To study the asymptotic behaviour of eigenvalues and eigenfunctions of a certain class of linear differential self adjoint operators, serious obstacles are found when we attempt to extend the semiclassical methods to a general multidimensional case. One may naturally suppose that the eigenfunctions (bound states) have bounded motions in the corresponding classical system as in their quantum counterparts. This suggests the investigation of invariant sets and their properties.
We see that some trajectories are organized in regular curves, being concentrated near periodic points. These are the so called KAM curves. “KAM” are the first letters of the names Kolmogorov (A.N.), Arnol’d (V.I.) and Moser (J.). They have discovered and proved the existence of the invariant tori. The author, V. F. Lazutkin, develops the study of KAM tori in the \(C^ \infty\) setting and eigenfunctions of the Schrödinger operator, which “tend” to KAM tori of the corresponding classical Hamiltonian system if the Planck constant tends to zero.
The book consists of two parts and an addendum by A. I. Shnirelman. The title of the addendum is “On the asymptotic properties of eigenfunctions in the regions of chaotic motion”. Also, it has 3 appendices; their titles are “Manifold” – brief expositions of geometry and analysis on manifolds, “Derivatives of super position” and “The stationary phase method”.
The title of Part I is “KAM theory”. It consists of four chapters.
Chapter I. Symplectic dynamical systems.
Symplectic manifold, symplectic dynamical systems, symplectic gluing, completely integrable Hamiltonian systems and systems in an annulus are explained.
Chapter II. KAM theorems.
KAM torus, KAM set, KAM theorems in an annulus are explained. Behaviours near a torus, near a periodic motion and near the boundary of planar convex billiards are explained.
Chapter III. Beyond the tori.
Pictures of stochastically near KAM tori are examined.
Chapter IV. Proof of the main theorem.
The title of Part II is “Eigenfunctions Asymptotics”. It consists of three chapters, chapter V, chapter VI and chapter VII. This part is the main part of the book, which consists of results by V. F. Lazutkin.
Chapter V. Laplace-Beltrami-Schrödinger operators and quasimodes.
Quasimodes and degenerated quasimodes are explained with the fundamental facts of LBS-operators.
Chapter VI. Maslov’s operator.
In this chapter, Maslov’s index theory and its application are explained. A global formula of Maslov’s operator is given.
Chapter VII. Quasimodes attached to a KAM set.
The canonical Maslov operator associated to a KAM set is studied.
Each chapter has a note, which contains source references.
The book consists of very interesting and very stimulative results. However, it is not easy to read through. There are several mistypes.

Reviewer: Y.Asoo (Okayama)

MSC:

58-02	Research exposition (monographs, survey articles) pertaining to global analysis
58J50	Spectral problems; spectral geometry; scattering theory on manifolds
37A30	Ergodic theorems, spectral theory, Markov operators
81Q20	Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
37Cxx	Smooth dynamical systems: general theory
37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70Hxx	Hamiltonian and Lagrangian mechanics
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35J10	Schrödinger operator, Schrödinger equation
35P20	Asymptotic distributions of eigenvalues in context of PDEs
35Q40	PDEs in connection with quantum mechanics

Keywords:

KAM tori; eigenfunctions; Schrödinger operator; symplectic dynamical systems