Derivation of the Gross-Pitaevskii Theory for Interacting Fermions in a Trap

TL;DR

This paper derives the Gross-Pitaevskii theory from the microscopic BCS energy functional for a dilute, trapped gas of interacting fermions with attractive interactions, establishing a rigorous link between microscopic and macroscopic descriptions.

Contribution

It provides a rigorous derivation of the Gross-Pitaevskii equation for interacting fermions in a trap from the BCS energy functional in the dilute limit.

Findings

Gross-Pitaevskii theory emerges as the first-order approximation of the BCS energy functional.

The derivation applies to a dilute gas of fermions with attractive interactions and a two-body bound state.

The results connect microscopic interactions with macroscopic quantum phenomena.

Abstract

We study a dilute gas of interacting fermions at temperature $T=0$ and chemical potential $\mu \in \mathbb{R}$. The particles are trapped by an external potential, and they interact via a microscopic attractive two-body potential with a two-body bound state. We prove the emergence of the macroscopic Gross-Pitaevskii theory as first-order contribution to the BCS energy functional in the regime of vanishing micro-to-macro scale parameter.