Andreev levels in a single-channel conductor

We calculate the subgap density of states of a disordered single-channel normal metal connected to a superconductor at one end (normal-metal-superconductor junction) or at both ends [superconductor-normal-metal-superconductor (SNS) junction]. The probability distribution of the energy of a bound state (Andreev level) is broadened by disorder. In the SNS case the twofold degeneracy of the Andreev levels is removed by disorder leading to a splitting in addition to the broadening. The distribution of the splitting is given precisely by Wigner's surmise from random-matrix theory. For strong disorder the mean density of states is largely unaffected by the proximity to the superconductor, because of localization, except in a narrow energy region near the Fermi level, where the density of states is suppressed with a log-normal tail.