The Bardeen-Cooper-Schrieffer functional of superconductivity and its mathematical properties (Q2795510)

The main purpose of this paper is to review several recent results concerning the mathematical properties of the Bardeen-Cooper-Schrieffer functional of superconductivity. The authors are mainly interested in the investigation of the critical temperature for a general class of interaction potentials, as well as a study of its dependence on external fields. It is also explained how the Ginzburg-Landau model can be derived from the Bardeen-Cooper-Schrieffer theory in a suitable parameter regime. In the final part of this paper it is established that that under a certain positivity assumption (which can be proved for a class of interaction potentials) and in the absence of external fields, the translation-invariant minimizer of the Bardeen-Cooper-Schrieffer functional is indeed a minimizer, namely that the translation symmetry is not broken. The proofs combine variational methods with a priori estimates.