KAM Theory and Semiclassical Approximations to Eigenfunctions

A complete exposition on the subject of KAM theory, including a section on Hamiltonian dynamics. The main results of the text concern the existence and persistence of KAM theory and the estimate of the measure of the set filled by KAM theory.

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This monograph contains a series of research papers on KAM theory. In the first, 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 semi-classical asymptotics to the eigenfunctions of the generalized Schroedinger operator. The main result is the asymptotic formulae for eigenfunctions and eigen values, using Maslov's operator, for the set of eigen values of positive density in the set of all eigenvalues. An addendum by A.I. Shnirelman treats eigenfunctions corresponding to the 'chaotic component' of the phase space.