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Register a new userThe BCS Gap Equation on a Banach Space Consisting of Functions both of the Temperature and of the Wave Vector
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Authors
Shuji Watanabe
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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.
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One of long-standing problems in mathematical studies of superconductivity is to show that the solution to the BCS gap equation is continuous in the temperature. In this paper we address this problem. We regard the BCS gap equation as a nonlinear integral equation on a Banach space consisting of continuous functions of both $T$ and $x$. Here, $T (geq 0)$ stands for the temperature and $x$ the kinetic energy of an electron minus the chemical potential. We show that the unique solution to the BCS gap equation obtained in our recent paper is continuous with respect to both $T$ and $x$ when $T$ is small enough. The proof is carried out based on the Banach fixed-point theorem.
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