Spectra of Random Matrices Close to Unitary and Scattering Theory for Discrete-Time Systems

We analyze statistical properties of complex eigenvalues of random matrices $\hat{A}$ close to unitary. Such matrices appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with discrete time. Deviation from unitarity are characterized by rank $M$ and eigenvalues $T_i, i=1,...,M$ of the matrix $\hat{T}=\hat{{\bf 1}}-\hat{A}^{\dagger}\hat{A}$. For the case M=1 we solve the problem completely by deriving the joint probability density of eigenvalues and calculating all $n-$ point correlation functions. For a general case we present the correlation function of secular determinants.