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Birman-Schwinger and the number of Andreev states in BCS superconductors

The number of bound states due to 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 the number of sub-gap states using the Birman-Schwinger principle. We show how to obtain upper bounds on the number of sub-gap states for small normal regions and derive 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.