Mathematical analysis of the gap function in the BCS model of superconductivity as a function of both the temperature and the wavevector

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We study the solution to the BCS gap equation in the BCS model of superconductivity. When the potential is a positive constant, we establish the existence and uniqueness of the solution so as to show smoothness of the solution on the basis of the implicit function theorem. When the potential is not a constant, we next show how the solution varies with the temperature on the basis of the Schauder fix-point theorem. Approximating the solution by a function of class C2 , we then show that the phase transition to a superconducting state is a second-order phase transition.