The BCS Functional for General Pair Interactions

TL;DR

This paper provides a rigorous mathematical analysis of the BCS functional for general pair interactions, establishing conditions for superconductivity and critical temperature behavior in fermionic gases.

Contribution

It extends the analysis of the BCS functional to general pair potentials and characterizes the existence of non-trivial solutions and critical temperature conditions.

Findings

Existence of non-trivial solutions is linked to negative eigenvalues of a linear operator.

A non-zero critical temperature exists for attractive potentials.

Critical temperature is exponentially small in the potential strength.

Abstract

The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.