  We study a $C^*$-dynamical system describing a particle coupled to an
infinitely extended heat bath at positive temperature. For small coupling
constant we prove return to equilibrium exponentially fast in time. The novelty
in this context is to model the particle by a harmonic or anharmonic
oscillator, respectively. The proof is based on explicit formulas for the time
evolution of Weyl operators in the harmonic oscillator case. In the anharmonic
oscillator case, a Dyson's expansion for the dynamics is essential. Moreover,
we show in the harmonic oscillator case, that $\R$ is the absolute continuous
spectrum of the Standard Liouvillean and that zero is a unique eigenvalue.
