Boundary conditions for Green's functions of spatially inhomogeneous superconducting systems

A system of semiclassical equations for the Green's functions of a superconductor with coincident arguments is proposed. The equations describe spatially inhomogeneous superconducting systems of the type of a tunnel junction and are distinguished by the fact that the Green's functions with different values of the indices labeling the left and right states do not mix with each other but satisfy independent equations. They also satisfy standard boundary conditions at infinity, i.e., the left-right and right-left functions tend at infinity to zero and the right-right and left-left functions to the values inherent in the massive superconductor. The Green's functions are discontinuous at the origin; matching conditions connecting all components with different values of the indices are obtained.