Andreev reflection and the semiclassical Bogoliuobov-De Gennes Hamiltonian

We present a semi-classical analysis of the opening of superchannels in gated mesoscopic SNS or SFS junctions. For perfect Josephson junctions (i.e. hardwall potential), this was considered by [2] 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. The problem of finding quantization rules for Andreev states near energy E close to the Fermi level, reduces to finding the zeroes of the determinant of a monodromy matrix, which we characterize partially by means of geometric quantities.