The uniqueness and approximation of a positive solution of the Bardeen-Cooper-Schrieffer gap equation

In this paper we study the Bardeen-Cooper-Schrieffer energy gap equation at finite temperatures. When the kernel is positive representing a phonon-dominant phase in a superconductor, the existence and uniqueness of a gap solution is established in a class which contains solutions obtainable from bounded domain approximations. The critical temperatures that characterize superconducting-normal phase transitions realized by bounded domain approximations and full space solutions are also investigated. It is shown under some sufficient conditions that these temperatures are identical. In this case the uniqueness of a full space solution follows directly. We will also present some examples for the nonuniqueness of solutions. The case of a kernel function with varying signs is also considered. It is shown that, at low temperatures, there exist nonzero gap solutions indicating a superconducting phase, while at high temperatures, the only solution is the zero solution, representing the dominance of the normal phase, which establishes again the existence of a transition temperature.