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Mathematical analysis of the multiband BCS gap equations in superconductivity. (English) Zbl 1062.82060
Summary: 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 \(T\to T_c\), 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.

82D55 Statistical mechanical studies of superconductors
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