由温度函数和波矢函数组成的Banach空间上的BCS间隙方程
The BCS Gap Equation on a Banach Space Consisting of Functions both of the Temperature and of the Wave Vector
摘要：
在以往的BCS超导能隙方程的数学研究中，能隙函数被视为仅由波矢函数组成的空间的一个元素。但我们把它看作是由温度和波矢函数组成的Banach空间的一个元素。本文首先利用隐函数定理证明了由BCS能隙方程得到的简化能隙方程存在关于温度的唯一的$C^2$类解。然后将BCS能隙方程视为上述Banach空间上的非线性积分方程，并利用Schauder不动点定理证明了BCS能隙方程存在唯一解.我们发现BCS能隙方程的解对于温度和波矢都是连续的，并且该解对于温度和波矢都是近似的$C^2$类函数。此外，BCS间隙方程的解在一定条件下退化为简化间隙方程的解。
英文摘要：
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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