Random scattering matrices for Andreev quantum dots with nonideal leads

We calculate the distribution of the scattering matrix at the Fermi level for chaotic normal-superconducting systems for the case of arbitrary coupling of the scattering region to the scattering channels. The derivation is based on the assumption of uniformly distributed scattering matrices at ideal coupling, which holds in the absence of a gap in the quasiparticle excitation spectrum. The resulting distribution is the analog of the Poisson kernel for the nonstandard symmetry classes introduced by Altland and Zirnbauer. We show that unlike the Poisson kernel, the analyticity-ergodicity constraint does not apply to our result. As a simple application, we calculate the distribution of the conductance for a single-channel chaotic Andreev quantum dot in a magnetic field.