Integrability and chaos in classical and quantum mechanics (Q1177074)

This paper deals with what the authors call Large Poincaré Systems (LPS). At the level of classical mechanics, LPS are characterized by Hamiltonians admitting a representation, in canonical coordinates \((J,\alpha)\), of the form \[ H(J,\alpha)=H_ 0(J)+\lambda V(J,\alpha), \] where \(V\) may be represented as a Fourier integral (not as a Fourier series!) with respect to \(\alpha=(\alpha_ 1,\dots,\alpha_ n)\). Thus, contrary to the familiar KAM theory, the \(\alpha_ i\) must not be viewed as angle coordinates. A characteristic feature of LPS is the occurrence of a continuum of resonances. The authors consider such systems classically as well as quantum mechanically and, in particular, in the context of quantum statistics. They propose a new integration method, in terms of series expansions in \(\lambda\), for the dynamics of density operators that uses an ``internal time'' and a non-unitary transformation induced thereby. One of the goals is to explain the second law of thermodynamics. The paper is written in a qualitative fashion; technical details are announced to appear in forthcoming articles.