Andreev reflection and the semiclassical Bogoliubov-de Gennes Hamiltonian: Resonant states

We present a semi-classical analysis of the opening of superchannels in gated mesoscopic SNS junctions. For perfect junctions (i.e. hard-wall potential), this was considered by [ChLeBl] in the framework of scattering matrices. Here we allow for imperfections in the junction, so that the complex order parameter continues as a smooth function, which is a constant in the superconducting banks, and vanishes rapidly inside the lead. We obtain quantization rules for resonant Andreev states near energy E close to the Fermi level, including the determination of the resonance width.