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

TL;DR

This paper introduces a new method based on the Birman-Schwinger principle to estimate the number of sub-gap states in BCS superconductors with inhomogeneities, providing bounds for different region sizes.

Contribution

It develops a novel approach using the Birman-Schwinger principle and inequalities to estimate bound states in inhomogeneous superconductors, extending previous methods.

Findings

Derived upper bounds for small normal regions

Estimated number of states for large regions using Szego theorem

Applicable to local inhomogeneities and external potentials

Abstract

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.