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.