Eigenfunctions of the Liouville operator, periodic orbits and the principle of uniformity

Abstract

We investigate the eigenvalue problem for the dynamical variables' evolution equation in classical mechanics, df dt = Lf where L is the Liouville operator, the generator of the unitary one-parameter group U t = e −Lt. We show that the non-constant eigenfunctions are distributions on the energy shell and nonvanishing on its elementary retracing invariant submanifolds: rational tori for the integrable case or periodic orbits for the chaotic case. The formalism unveils an equivalent statement, concerning the definition of a measure on the Hilbert space of dynamical variables, for the principle of uniformity. Introducing this measure, which is delta concentrated on the periodic orbits, we are able to derive the classical sum rules obtained from the principle of uniformity from the way the periodic orbits proliferate for increasing periods.

Projective representation of the Galilei group for classical and quantum–classical systems*

“…It will be shown how to associate a Hilbert space to the classical phase space, the inner product in this Hilbert space, and the equation of motion of the state vector (or wave functions). Of special interest will be the form of the KvN waves when angle-action variables are used [25,26,27]. It will be shown, using formulas from quantum mechanics, that the geometric phase factor acquired by the KvN states is related to the Hannay angle.…”

de Aguiar and de Carvalho Reply:

“…according to the classical sum rule of Hannay and Ozorio de Almeida [5,6]. On the other hand, for v fi 2pn͞T k , we may approximate 1͞j det͑M The sum in brackets in now convergent, yielding a finite weight to all v 0 s where the two spectra differ.…”