The dilute Fermi gas via Bogoliubov theory

TL;DR

This paper presents a new derivation of the ground state energy of dilute three-dimensional Fermi gases using Bogoliubov theory, improving error estimates and extending applicability to more regular potentials.

Contribution

It introduces a Bogoliubov-inspired method to derive the ground state energy formula, enhancing previous results with better error bounds and broader potential classes.

Findings

Derived the ground state energy formula using a new Bogoliubov-based approach.

Extended applicability to more regular interaction potentials.

Provided improved error estimates on the energy asymptotics.

Abstract

We study the ground state properties of interacting Fermi gases in the dilute regime, in three dimensions. We compute the ground state energy of the system, for positive interaction potentials. We recover a well-known expression for the ground state energy at second order in the particle density, which depends on the interaction potential only via its scattering length. The first proof of this result has been given by Lieb, Seiringer and Solovej. In this paper we give a new derivation of this formula, using a different method; it is inspired by Bogoliubov theory, and it makes use of the almost-bosonic nature of the low-energy excitations of the systems. With respect to previous work, our result applies to a more regular class of interaction potentials, but it comes with improved error estimates on the ground state energy asymptotics in the density.