At low temperature, the concentration of quasiparticles observed in superconducting circuits far exceeds the predictions of microscopic BCS theory at equilibrium. As a source of dissipation, these excess quasiparticles degrade the performance of various devices. Therefore, understanding their dynamics, especially their recombination into Cooper pairs, is an active topic of current research. In disordered superconductors, spatial fluctuations in the superconducting gap can trap quasiparticles and modify their eigenspectrum. Since this spectrum plays a key role in quasiparticle dynamics, it must be carefully investigated. To this end, we introduce a toy model of a single local gap suppression. Specifically, we consider a shallow disk-shaped gap variation in a clean superconductor. Using a semiclassical approximation, we demonstrate the existence of multiple bound states and give the dependence of their number on the size and depth of the gap suppression. Extending our analysis beyond the semiclassical regime, in dimensions larger than one, we observe an infinite number of bound states very close to the gap edge, even for an arbitrarily small gap variation. These results deepen our understanding of trapped quasiparticles and may have important implications for their recombination in disordered superconductors.

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• Andreev bound states
• Superconducting gap
• Superconductivity
• Bogoliubov-de Gennes equations
• Semiclassical methods