Smoothness and monotone decreasingness of the solution to the BCS-Bogoliubov gap equation for superconductivity

TL;DR

This paper proves that the solution to the BCS-Bogoliubov gap equation for superconductivity is smooth, continuous, and monotone decreasing with respect to temperature, providing detailed regularity properties.

Contribution

It establishes the smoothness, monotonicity, and differentiability properties of the solution to the BCS gap equation with respect to temperature.

Findings

Solution is continuous in temperature and energy.

Solution is Lipschitz continuous and monotone decreasing in temperature.

Second derivatives of the solution are continuous, indicating smoothness.

Abstract

We show the temperature dependence such as smoothness and monotone decreasingness with respect to the temperature of the solution to the BCS-Bogoliubov gap equation for superconductivity. Here the temperature belongs to the closed interval $[0,\, \tau]$ with $\tau>0$ nearly equal to half of the transition temperature. We show that the solution is continuous with respect to both the temperature and the energy, and that the solution is Lipschitz continuous and monotone decreasing with respect to the temperature. Moreover, we show that the solution is partially differentiable with respect to the temperature twice and the second-order partial derivative is continuous with respect to both the temperature and the energy, or that the solution is approximated by such a smooth function.