Asymptotic stability of a wide class of stationary solutions for the Hartree and Schr\"{o}dinger equations for infinitely many particles

TL;DR

This paper proves the asymptotic stability of a broad class of stationary solutions, including the zero-temperature Fermi gas, for the Hartree and Schrödinger equations describing infinitely many interacting fermions.

Contribution

It establishes the asymptotic stability of stationary solutions for these equations, extending previous results to include physically significant states like the zero-temperature Fermi gas.

Findings

Proved asymptotic stability for a wide class of stationary solutions.

Included the physically important Fermi gas at zero temperature.

Extended prior stability results to more general states.

Abstract

We consider the Hartree and Schr\"{o}dinger equations describing the time evolution of wave functions of infinitely many interacting fermions in three-dimensional space. These equations can be formulated using density operators, and they have infinitely many stationary solutions. In this paper, we prove the asymptotic stability of a wide class of stationary solutions. We emphasize that our result includes Fermi gas at zero temperature. This is one of the most important steady states from the physics point of view; however, its asymptotic stability has been left open after Lewin and Sabin first formulated this stability problem and gave significant results in their seminal work [arXiv:1310.0604].