Physica 80A (1975) 323-338 ~) North-Holland Publishing Co. SPECTRAL PROPERTIES OF LIOUVILLE OPERATORS AND THEIR PHYSICAL INTERPRETATION H. SPOHN Fachbereieh Physik der Ludwig-Maximilians-Universitdt Mgnchen, 8 Munich 2, Germany Received 7 February 1975 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 be- longing to a separable hamiltonian system in terms of the transformed hamiltonian function. In the general case we prove the symmetry of the spectrum and a structure theorem about the point spectrum. 1. Introduction It has already been known for a long time 1) that the fundamental evolution equation of classical statistical mechanics, the Liouville equation i --:--Q(t) = i [H, ~1, (1.1) 0t becomes formally identical to the Schr6dinger equation, if one considers i [H, ] as a linear operator in the Hilbert space Aa2(F). ([, ] is the Poisson bracket. 1~ is the phase space, H the hamiltonian function and i [H, ] the Liouville operator of the system considered.) This observation gave rise to a large number of investi- gations, which use mathematical techniques originally developed in the context of quantum mechanics as, e.g., projection operators, perturbation expansions for self-adjoint operators, diagram techniques for resummation etc.; techniques which are by now standard in kinetic theory2). Thus the fact that the Liouville operator i [H, ] is a self-adjoint operator in ,~2(/n) has been fully exploited. How- ever, quite contrary to the case of hamiltonian operators, the spectral properties of Liouville operators and their physical interpretation have not been investi- 323