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

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.