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

TL;DR

This paper investigates the asymptotic behavior of the Hartree equation near steady states, establishing phase-mixing estimates that demonstrate density decay and quantum Landau damping uniformly in the semiclassical limit.

Contribution

It provides the first quantum analogue of Landau damping with uniform semiclassical estimates for short-range interactions satisfying Penrose stability.

Findings

Density decay and scattering in weighted quantum Sobolev spaces

Quantum Landau damping established for the Hartree equation

Results hold uniformly 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.