The BCS Gap Equation on a Banach Space Consisting of Functions both of the Temperature and of the Wave Vector

TL;DR

This paper extends the mathematical analysis of the BCS gap equation by considering a Banach space of functions depending on both temperature and wave vector, proving existence, uniqueness, and regularity of solutions.

Contribution

It introduces a Banach space framework for the BCS gap equation and establishes the existence and smoothness of solutions using the implicit function theorem and Schauder fixed-point theorem.

Findings

Unique $C^2$ solution with respect to temperature

Solution is continuous in both temperature and wave vector

Solution approximates a $C^2$ function in both variables

Abstract

In previous mathematical studies of the BCS gap equation of superconductivity, the gap function was regarded as an element of a space consisting of functions of the wave vector only. But we regard it as an element of a Banach space consisting of functions both of the temperature and of the wave vector. On the basis of the implicit function theorem we first show that there is a unique solution of class $C^2$ with respect to the temperature, to the simplified gap equation obtained from the BCS gap equation. We then regard the BCS gap equation as a nonlinear integral equation on the Banach space above, and show that there is a unique solution to the BCS gap equation on the basis of the Schauder fixed-point theorem. We find that the solution to the BCS gap equation is continuous with respect to both the temperature and the wave vector, and that the solution is approximated by a function of class $C^2$ with respect to both the temperature and the wave vector. Moreover, the solution to the BCS gap equation is shown to reduce to the solution to the simplified gap equation under a certain condition.