
VOLUME 72 (2000)  ISSUE 8 
PAGE 605

Spectra of Random Contractions and Scattering Theory for DiscreteTime Systems
Fyodorov Y.V., Sommers H.J.
PACS: 03.65.Nk, 05.45.Mt
Random contractions (subunitary 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 Ν χ N random matrices A of such a type, corresponding to systems with broken timereversal invariance. Deviations from unitarity are characterized by rank Af < N and a set of eigenvalues 0 < T* < 1, i — 1,..., Μ of the matrix Τ = 1 — A*A. We solve the problem completely by deriving the joint probability density of N complex eigenvalues and calculating all ηpoint correlation functions. In the limit Ν » M,n the correlation functions acquire the universal form found earlier for weakly nonHermitian random matrices.

