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× N symmetric random matrices of the form H=Ĥ-iΓ̂, where both Ĥ and Γ̂ are real symmetric, Ĥ is random Gaussian and Γ̂ is such that NTr Γ̂²₂~ Tr Ĥ₁² when N→ ∞. When Γ̂≥ 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 χ² distribution to the case of overlapping resonances. We also consider a different class of "almost real" matrices when Γ̂ is random and uncorrelated with Ĥ.