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

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Abstract
We investigate the eigenvalue problem for the dynamical variables' evolution equation in classical mechanics, where is the Liouville operator, the generator of the unitary one-parameter group . We show that the non-constant eigenfunctions are distributions on the energy shell and non-vanishing 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.

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