Density of states in a mesoscopic SNS junction

Semiclassical theory of proximity effect predicts a gap $E_{g} \sim \hbar D/L^2$ in the excitation spectrum of a long diffusive superconductor/normal-metal/superconductor (SNS) junction. Mesoscopic fluctuations lead to anomalously localized states in the normal part of the junction. As a result, a non-zero, yet exponentially small, density of states (DOS) appears at energies below $E_{g}$. In the framework of the supermatrix nonlinear $\sigma$-model these prelocalized states are due to instanton configurations with broken supersymmetry. The exact result for the DOS near the semiclassical threshold is found provided the dimensionless conductance of the normal part $G_{\rm N}$ is large. The case of poorly transparent interfaces between the normal and superconductive regions is also considered. In this limit the total number of the subgap states may be large.