# Is the solution to the BCS gap equation continuous in the temperature - Mathematical Physics

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

Autor: Shuji Watanabe

Fuente: https://arxiv.org/