Birman-Schwinger and the number of Andreev states in Bardeen-Cooper-Schrieffer superconductors

Abstract

The number of bound states resulting from inhomogeneities in a BCS superconductor is usually established either by variational means or via exact solutions of particularly simple, symmetric perturbations. Here, we propose estimating subgap states using the Birman-Schwinger principle. We derive upper bounds on the number of subgap states for small normal regions and find a suitable Cwikel-Lieb-Rozenblum inequality. We also estimate the number of such states for large normal regions using high dimensional generalizations of the Szego theorem. The method works equally well for local inhomogeneities of the order parameter and for external potentials.

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