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The Wigner-Fokker-Planck equation: stationary states and large time behavior. (English) Zbl 1253.82065

Summary: 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.

MSC:

82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
35S10 Initial value problems for PDEs with pseudodifferential operators
74H40 Long-time behavior of solutions for dynamical problems in solid mechanics
81Q15 Perturbation theories for operators and differential equations in quantum theory
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