Localized spectral asymptotics for boundary value problems and correlation effects in the free Fermi gas in general domains

TL;DR

This paper derives explicit formulas for the pair correlation function of the free Fermi gas in various geometries, confirming the local density approximation and revealing universal correlation effects in the thermodynamic limit.

Contribution

It provides rigorous spectral asymptotics and explicit correlation formulas for the free Fermi gas in general domains, extending previous results to arbitrary geometries and boundary conditions.

Findings

Explicit pair correlation formulas valid for general geometries.

Asymptotic validity of the local density approximation for exchange energy.

Universal correlation effects independent of domain shape.

Abstract

We rigorously derive explicit formulae for the pair correlation function of the ground state of the free Fermi gas in the thermodynamic limit for general geometries of the macroscopic regions occupied by the particles and arbitrary dimension. As a consequence we also establish the asymptotic validity of the local density approximation for the corresponding exchange energy. At constant density these formulae are universal and do not depend on the geometry of the underlying macroscopic domain. In order to identify the correlation effects in the thermodynamic limit, we prove a local Weyl law for the spectral asymptotics of the Laplacian for certain quantum observables which are themselves dependent on a small parameter under very general boundary conditions.