Is the solution to the BCS gap equation continuous in the temperature ?

TL;DR

This paper proves that the solution to the BCS gap equation is continuous in temperature and energy for small temperatures, addressing a longstanding mathematical problem in superconductivity theory.

Contribution

It demonstrates the continuity of the BCS gap equation solution in both temperature and energy using Banach fixed-point theorem, for small temperatures.

Findings

Solution is continuous in T and x for small T.

Uses Banach fixed-point theorem for proof.

Addresses a long-standing mathematical problem.

Abstract

One of long-standing problems in mathematical studies of superconductivity is to show that the solution to the BCS gap equation is continuous in the temperature. In this paper we address this problem. We regard the BCS gap equation as a nonlinear integral equation on a Banach space consisting of continuous functions of both $T$ and $x$. Here, $T (\geq 0)$ stands for the temperature and $x$ the kinetic energy of an electron minus the chemical potential. We show that the unique solution to the BCS gap equation obtained in our recent paper is continuous with respect to both $T$ and $x$ when $T$ is small enough. The proof is carried out based on the Banach fixed-point theorem.