Statistics of S-matrix poles for chaotic systems with broken time reversal invariance: a conjecture

TL;DR

This paper proposes a conjecture for the statistical distribution of S-matrix poles in chaotic systems lacking time-reversal symmetry, linking random matrix theory with quantum scattering properties and validating it against known delay statistics.

Contribution

It introduces a new conjecture for the eigenvalue statistics of non-Hermitian random matrices in broken time-reversal systems and extends the understanding of complex eigenvalue densities.

Findings

Reproduces known Wigner time delay statistics

Derives spectral form factor and number variance

Finds density of complex eigenvalues for real asymmetric matrices

Abstract

In the framework of a random matrix description of chaotic quantum scattering the positions of $S-$matrix poles are given by complex eigenvalues $Z_i$ of an effective non-Hermitian random-matrix Hamiltonian. We put forward a conjecture on statistics of $Z_i$ for systems with broken time-reversal invariance and verify that it allows to reproduce statistical characteristics of Wigner time delays known from independent calculations. We analyze the ensuing two-point statistical measures as e.g. spectral form factor and the number variance. In addition we find the density of complex eigenvalues of real asymmetric matrices generalizing the recent result by Efetov\cite{Efnh}.