Spectrum and thermodynamic currents in one dimensional Josephson   elements
  The dc Josephson effect is considered from the thermodynamic point of view. Universal thermodynamic equations, relating both bound and continuum contributions to the Josephson current with the normal electron scattering amplitudes are derived for the single mode case. To derive these equations we use and further develop the method of spatial separation between the superconducting and normal parts of the junction. We also use this method to find the Andreev bound states in structures containing superconducting components. The general thermodynamic formulas are applied to the calculation of the current in various Josephson-type structures. In particular, the crucial role of the continuum contribution is demonstrated, even for short junctions (where it is usually neglected). We also find structures where the bound states supporting the giant currents are well separated; thus they can, hopefully, be populated nonuniformly and such current can be measured. 
A. Krichevsky, M. Schechter, Y. Imry, and Y. Levinson