Andreev reflection, Andreev states, and long ballistic SNS junction

The analysis in the present paper is based on the most known concept introduced by the brilliant physicist Alexander Andreev: Andreev bound states in a normal metal sandwiched between two superconductors. The paper presents results of direct calculations of {\em ab initio} expressions for the currents in a long ballistic SNS junction. The expressions are expanded in $1/L$ ($L$ is the thickness of the normal layer). The main contribution $\propto 1/L$ to the current agrees with the results obtained in the past, but the analysis suggests a new physical picture of the charge transport through the junction free from the problem with the charge conservation law. The saw-tooth current-phase relation at $T=0$ directly follows from the Galilean invariance of the Bogolyubov-de Gennes equations proved in the paper. The proof is valid for any variation of the energy gap in space if the Andreev reflection is the only scattering process. The respective roles of the contributions of bound and continuum states to the current are clarified. They depend on the junction dimensionality.