The quantitative semi-classical limit of a large Fermi system at zero temperature

TL;DR

This paper analyzes the convergence of a large fermionic system's quantum state to classical Thomas-Fermi theory in a semi-classical limit, providing quantitative convergence rates including Coulomb interactions.

Contribution

It introduces a novel quantification of the convergence rate of the Wigner function to Thomas-Fermi theory in a semi-classical limit for large fermion systems.

Findings

Established convergence of Wigner function to Thomas-Fermi ground state

Quantified convergence rate with respect to semi-classical parameter

Included Coulomb and singular potentials in the analysis

Abstract

In this article we consider a large system of fermions in a combined mean-field and semiclassical limit, in three dimensions. We investigate the convergence of the Wigner function of the ground state, towards the classical Thomas-Fermi theory. The main novelty of the present article is quantifying the convergence rate with respect to the semi-classical parameter. One of the main ingredients is a recent result on the validity of semi-classical commutator estimates satisfied by the Hartree theory. Singular potentials, up to the Coulomb interaction, are included.