Condensation of fermion pairs in a domain

TL;DR

This paper derives a macroscopic Gross-Pitaevskii theory for fermion pair condensation in a bounded domain from microscopic BCS theory, establishing boundary condition consistency and domain approximation continuity.

Contribution

It provides a rigorous derivation of the GP theory for fermion pairs with Dirichlet boundary conditions from microscopic BCS theory.

Findings

Derivation of macroscopic GP theory from microscopic BCS theory.

Proof of continuity of GP energy under domain approximations.

Establishment of Dirichlet boundary conditions in the effective theory.

Abstract

We consider a gas of fermions at zero temperature and low density, interacting via a microscopic two body potential which admits a bound state. The particles are confined to a domain with Dirichlet (i.e. zero) boundary conditions. Starting from the microscopic BCS theory, we derive an effective macroscopic Gross-Pitaevskii (GP) theory describing the condensate of fermion pairs. The GP theory also has Dirichlet boundary conditions. Along the way, we prove that the GP energy, defined with Dirichlet boundary conditions on a bounded Lipschitz domain, is continuous under interior and exterior approximations of that domain.