Mathematical analysis of the multiband BCS gap equations in superconductivity

In this paper, we present a mathematical analysis for the phonon-dominated multiband isotropic and anisotropic BCS gap equations at any finite temperature T. We establish the existence of a critical temperature T c so that, when T < T c , there exists a unique positive gap solution, representing the superconducting phase; when T > T c , the only nonnegative gap solution is the zero solution, representing the normal phase. Furthermore, when T = T c , we prove that the only gap solution is the zero solution and that the positive gap solution depend on the temperature T < T c monotonically and continuously. In particular, as T → T c , the gap solution tends to zero, which enables us to determine the critical temperature T c . In the isotropic case where the entries of the interaction matrix K are all constants, we are able to derive an elegant T c equation which says that T c depends only on the largest positive eigenvalue of K but does not depend on the other details of K. In the anisotropic case, we may derive a similar T c equation in the context of the Markowitz–Kadanoff model and we prove that the presence of anisotropic fluctuations enhances T c as in the single-band case. A special consequence of these results is that the half-unity exponent isotope effect may rigorously be proved in the multiband BCS theory, isotropic or anisotropic.