Stability of homogeneous equilibria of the Hartree-Fock equation, for its equivalent formulation for random fields

TL;DR

This paper proves the asymptotic stability of homogeneous equilibrium states in the Hartree-Fock equation, including the exchange term, using a random fields framework, and shows perturbations scatter to linear waves.

Contribution

It is the first to analyze the full Hartree-Fock equation with exchange term for stability of homogeneous states in a random fields setting.

Findings

Homogeneous equilibria are asymptotically stable in large dimensions.

Perturbations to these states scatter to linear waves.

The exchange term is incorporated into the stability analysis for the first time.

Abstract

The Hartree-Fock equation admits homogeneous states that model infinitely many particles at equilibrium. We prove their asymptotic stability in large dimensions, under assumptions on the linearised operator. Perturbations are moreover showed to scatter to linear waves. We obtain this result for the equivalent formulation of the Hartree-Fock equation in the framework of random fields. The main novelty is to study the full Hartree-Fock equation, including for the first time the exchange term in the study of these stationary solutions.