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 $\hat{T}=\hat{\bf 1}-\hat{A}^{\dagger}\hat{A}$. 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.
- Publication:
-
Soviet Journal of Experimental and Theoretical Physics Letters
- Pub Date:
- October 2000
- DOI:
- 10.1134/1.1335121
- arXiv:
- arXiv:nlin/0005061
- Bibcode:
- 2000JETPL..72..422F
- Keywords:
-
- Nonlinear Sciences - Chaotic Dynamics;
- Condensed Matter
- E-Print:
- Complete solution of the problem discussed in earlier e-preprint nlin.CD/0002034