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 Chtchelkatchev, Lesovik and Blatter 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. 
A.Bensouissi, N.M'hadbi, M.Rouleux
math-ph math.MP