S-matrix poles for chaotic quantum systems as eigenvalues of complex symmetric random matrices: from isolated to overlapping resonances

Abstract

We study complex eigenvalues of large $N\times N$ symmetric random matrices of the form ${\cal H}=\hat{H}-i\hat{\Gamma}$, where both $\hat{H}$ and $\hat{\Gamma}$ are real symmetric, $\hat{H}$ is random Gaussian and $\hat{\Gamma}$ is such that $NTr \hat{\Gamma}^2_2\sim Tr \hat{H}_1^2$ when $N\to \infty$. When $\hat{\Gamma}\ge 0$ the model can be used to describe the universal statistics of S-matrix poles (resonances) in the complex energy plane. We derive the ensuing distribution of the resonance widths which generalizes the well-known $\chi^2$ distribution to the case of overlapping resonances. We also consider a different class of "almost real" matrices when $\hat{\Gamma}$ is random and uncorrelated with $\hat{H}$.