An optimal upper bound for the dilute Fermi gas in three dimensions

TL;DR

This paper rigorously derives an optimal upper bound for the correlation energy of dilute three-dimensional Fermi gases, improving previous bounds with precise error estimates in the thermodynamic limit.

Contribution

It provides the first rigorous first-order upper bound with an optimal error term for the correlation energy of dilute 3D Fermi gases.

Findings

Established an upper bound with error  in the dilute regime.

Improved the lower bound estimate with an error .

Enhanced understanding of correlation energy behavior in dilute Fermi systems.

Abstract

In a system of interacting fermions, the correlation energy is defined as the difference between the energy of the ground state and the one of the free Fermi gas. We consider $N$ interacting spin $1/2$ fermions in the dilute regime, i.e., $\rho\ll 1$ where $\rho$ is the total density of the system. We rigorously derive a first order upper bound for the correlation energy with an optimal error term of the order $\mathcal{O}(\rho^{7/3})$ in the thermodynamic limit. Moreover, we improve the lower bound estimate with respect to previous results getting an error $\mathcal{O}(\rho^{2+1/5})$.