Andreev States in long shallow SNS constrictions

We study Andreev bound states in a long shallow normal constriction, which is open to a superconductor at both ends. The interesting features of such setup include the absence of electron-hole symmetry and the interference of electron waves along the constriction. We compare results of a numerical approach based on the Bogoliubov equations with those of a refined semiclassical description. Three types of Andreev bound states occur in the constriction: i) one where both electron and hole wave part of the bound state propagate through the constriction, ii) one where neither electron nor hole wave part propagate, and iii) one where only the electron wave propagates. We show that in a wide energy region the spacing between the Andreev states is strongly modulated by the interference of electron waves passing the constriction. Typeset using REVTEX 1