The Josephson current through a long quantum wire

The dc Josephson current through a long SNS junction receives contributions from both Andreev bound states localized in the normal region and scattering states incoming from the superconducting leads. We show that in the limit of a long junction, this current, at low temperatures, can be expressed entirely in terms of properties of the Andreev bound states at the Fermi energy: the normal and Andreev reflection amplitudes at the left- and right-hand S-N interfaces. This has important implications for treating interactions in such systems.