Spectra of Random Contractions and Scattering Theory for Discrete-Time Systems

Abstract

Random contractions (sub-unitary random matrices) appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with discrete time. We analyze statistical properties of complex eigenvalues of generic $N\times N$ random matrices $\hat{A}$ of such a type, corresponding to systems with broken time-reversal invariance. Deviations from unitarity are characterized by rank $M\le N$ and a set of eigenvalues $0<T_i\le 1,i=1,...,M$ of the matrix $\hat{T}=\hat{1}-{\hat{A}}^{†}\hat{A}$. We solve the problem completely by deriving the joint probability density of $N$ complex eigenvalues and calculating all $n-$ point correlation functions. In the limit $N>>M,n$ the correlation functions acquire the universal form found earlier for weakly non-Hermitian random matrices.

Complete solution of the problem discussed in earlier e-preprint nlin.CD/0002034

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