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Abstract and Applied Analysis
Volume 2013, Article ID 932085, 5 pages
Research Article

The Solution to the BCS Gap Equation for Superconductivity and Its Temperature Dependence

Division of Mathematical Sciences, Graduate School of Engineering, Gunma University, 4-2 Aramaki-machi, Maebashi 371-8510, Japan

Received 27 May 2013; Accepted 9 August 2013

Academic Editor: Santanu Saha Ray

Copyright © 2013 Shuji Watanabe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


From the viewpoint of operator theory, we deal with the temperature dependence of the solution to the BCS gap equation for superconductivity. When the potential is a positive constant, the BCS gap equation reduces to the simple gap equation. We first show that there is a unique nonnegative solution to the simple gap equation, that it is continuous and strictly decreasing, and that it is of class with respect to the temperature. We next deal with the case where the potential is not a constant but a function. When the potential is not a constant, we give another proof of the existence and uniqueness of the solution to the BCS gap equation, and show how the solution varies with the temperature. We finally show that the solution to the BCS gap equation is indeed continuous with respect to both the temperature and the energy under a certain condition when the potential is not a constant.