Statistics of S-matrix poles in Few-Channel Chaotic Scattering: Crossover from Isolated to Overlapping Resonances

We derive the explicit expression for the distribution of resonance widths in a chaotic quantum system coupled to continua via M equivalent open channels. It describes a crossover from the $\chi^2$ distribution (regime of isolated resonances) to a broad power-like distribution typical for the regime of overlapping resonances. The first moment is found to reproduce exactly the Moldauer-Simonius relation between the mean resonance width and the transmission coefficient. This fact may serve as another manifestation of equivalence between the spectral and the ensemble averaging.