On the ground state energy of the delta-function Fermi gas

TL;DR

This paper rigorously establishes the weak coupling asymptotics of the ground state energy for the delta-function Fermi gas, providing a solid mathematical foundation for results previously derived heuristically.

Contribution

It makes rigorous the weak coupling asymptotics of the ground state energy of the delta-function Fermi gas, previously obtained heuristically, and introduces a method applicable to related integral equations.

Findings

Rigorous proof of weak coupling asymptotics to order γ

Method applicable to Gaudin integral equation

Potential for computing further asymptotics

Abstract

The weak coupling asymptotics to order $\gamma$ of the ground state energy of the delta-function Fermi gas, derived heuristically in the literature, is here made rigorous. Further asymptotics are in principle computable. The analysis applies to the Gaudin integral equation a method previously used by one of the authors for the asymptotics of large Toeplitz matrices.