Random matrix theory of the proximity effect in disordered wires

We study analytically the local density of states in a disordered normal-metal wire (N) at ballistic distance to a superconductor $\mathrm{}\left(S\right).$ Our calculation is based on a scattering-matrix approach, which concerns for wave-function localization in the normal metal, and extends beyond the conventional semiclassical theory based on Usadel and Eilenberger equations. We also analyze how a finite transparency of the $\mathrm{NS}$ interface modifies the spectral proximity effect and demonstrate that our results agree in the dirty diffusive limit with those obtained from the Usadel equation.