  We consider the linear Wigner-Fokker-Planck equation subject to confining
potentials which are smooth perturbations of the harmonic oscillator potential.
For a certain class of perturbations we prove that the equation admits a unique
stationary solution in a weighted Sobolev space. A key ingredient of the proof
is a new result on the existence of spectral gaps for Fokker-Planck type
operators in certain weighted $L^2$-spaces. In addition we show that the steady
state corresponds to a positive density matrix operator with unit trace and
that the solutions of the time-dependent problem converge towards the steady
state with an exponential rate.
