Phase mixing for the Hartree equation and Landau damping in the semiclassical limit

Abstract

The asymptotic behaviour of the Hartree equation is studied near translation-invariant steady states. For short-range interaction kernels satisfying a uniform Penrose stability condition, including the screened Coulomb interaction, phase-mixing estimates in finite regularity are established. These demonstrate density decay and scattering of solutions in weighted quantum Sobolev spaces, providing a quantum analogue of Landau damping in classical plasma physics. The results hold uniformly in the semiclassical limit, thereby bridging the quantum and classical regimes.

arXiv commentRevised version. Added a section identifying the semiclassical limit of Hartree scattering profiles with Vlasov scattering profiles, conditional on convergence at each fixed time. The main theorem now explicitly states global existence and uniqueness. Regularity assumptions have been strengthened, and the sufficient Penrose criterion, normalisations and several estimates have been corrected