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

We study complex eigenvalues of large $N\times N$ symmetric random matrices of the form $\mathcal{H}=\widehat{H}-i\widehat{\mathrm{\Gamma }}$, where both $\widehat{H}$ and $\widehat{\mathrm{\Gamma }}$ are real symmetric, $\widehat{H}$ is random Gaussian and $\widehat{\mathrm{\Gamma }}$ is such that $NTr{\widehat{\mathrm{\Gamma }}}_2^2\sim Tr{\widehat{H}}_1^2$ when $N\to \mathrm{\infty }$. When $\widehat{\mathrm{\Gamma }}\ge 0$ 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 $\widehat{\mathrm{\Gamma }}$ is random and uncorrelated with $\widehat{H}$.