The Wigner-Fokker-Planck equation: Stationary states and large-time behavior

Details The Wigner-Fokker-Planck equation: Stationary states and large-time behavior The Wigner-Fokker-Planck equation: Stationary states and large-time behavior

2010-10-13

arxiv.org/abs/1010.2791v3

Mathematics

Analysis of PDEs

This manuscript essentially replaces (and corrects a mistake found in) the submission arXiv:0707.2445, by establishing a new f

Scientific paper

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.

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