On the Bardeen-Cooper-Schrieffer integral equation in the theory of superconductivity

The Bardeen-Cooper-Schrieffer integral equation with a positive kernel is studied in full generality. It is shown that, there exists a unique finite transition temperature, T _c so that, if T _c ,the equation possesses a positive solution, representing the onset of the superconducting phase, while if T>T _c ,the only solution of the equation is the trivial one, indicating the occurrence of the normal phase. Moreover, it is demonstrated that such a positive solution may be approximated by a sequence of solutions of the equation restricted on bounded domains. This latter result provides a useful computational scheme for the problem.