Spectral properties of Liouville operators and their physical interpretation

The eigenvalue zero of a Liouville operator determines the bound and scattering states whereas the spectrum outside zero determines the approach to equilibrium of the hamiltonian system belonging to it. We completely characterize the spectral properties of the Liouville operator belonging to a separable hamiltonian system in terms of the transformed hamiltonian functions. In the general case we prove the symmetry of the spectrum and a structure theorem about the point spectrum.