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 $\mathcal{H}=\hat{H}-i\hat{\Gamma }$, 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 $\hat{H}$.

8 pages+2 eps figures

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