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posted on 15.11.2013by Anton Arnold, Irene M. Gamba, Maria Pia Gualdani, Stephane Mischler, Clement Mouhot, Christof Sparber
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.
FWF (project \Quantum Transport Equations:
Kinetic, Relativistic, and Di usive Phenomena" and Wissenschaftskolleg \Di erentialgleichungen"),
the OAD (Amadeus project), and the Newton Institute of Cambridge University. I. M.
Gamba is supported by NSF-DMS 0807712. M. P. Gualdani is supported by NSF-DMS-1109682. Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and
Technology (KAUST). C. Sparber has been supported by the Royal Society through his Royal
Society University Research Fellowship. Support from the Institute of Computational Engineering
and Sciences at the University of Texas at Austin is also gratefully acknowledged.