EIGENFUNCTIONS OF THE LIOUVILLE OPERATOR, PERIODIC-ORBITS AND THE PRINCIPLE OF UNIFORMITY

We investigate the eigenvalue problem for the dynamical variables' evo lution equation in classical mechanics df/dt = Lf where L is the Liouv ille 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 non-vanishing on its elementary retracing inv ariant submanifolds: rational tori for the integrable case or periodic orbits for the chaotic case. The formalism unveils an equivalent stat ement, 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 a ble to derive the classical sum rules obtained from the principle of u niformity from the way the periodic orbits proliferate for increasing periods.