Smoothness of the Gap Function in the BCS-Bogoliubov Theory of Superconductivity

TL;DR

This paper proves that the squared gap function in the BCS-Bogoliubov theory is twice continuously differentiable on the temperature interval, decreases monotonically, and describes its behavior at the critical temperature.

Contribution

It establishes the smoothness and monotonicity of the gap function in the BCS theory, providing rigorous mathematical properties of the gap function.

Findings

Squared gap function is C^2 on [0, Tc]

Gap function is monotonically decreasing on [0, Tc]

Behavior of the gap function at Tc is characterized

Abstract

We deal with the gap equation in the BCS-Bogoliubov theory of superconductivity, where the gap function is a function of the temperature $T$ only. We show that the squared gap function is of class $C^2$ on the closed interval $[\,0,\,T_c\,]$. Here, $T_c$ stands for the transition temperature. Furthermore, we show that the gap function is monotonically decreasing on $[0,\,T_c]$ and obtain the behavior of the gap function at $T=T_c$. We mathematically point out some more properties of the gap function.