  We analyze the long time behavior of transport equations for a class of
dissipative quantum systems with Fokker-planck type scattering operator,
subject to confining potentials of harmonic oscillator type. We establish the
conditions under which there exists a thermal equilibrium state and prove
exponential decay towards it, using (classical) entropy-methods. Additionally,
we give precise dispersion estimates in the cases were no equilibrium state
exists.
