Super-sharp resonances in chaotic wave scattering

Wave scattering in chaotic systems can be characterized by its spectrum of resonances, $z_n=E_n-i\frac{{\mathrm{\Gamma }}_n}{2}$, where $E_n$ is related to the energy and ${\mathrm{\Gamma }}_n$ is the decay rate or width of the resonance. If the corresponding ray dynamics is chaotic, a gap is believed to develop in the large-energy limit: almost all ${\mathrm{\Gamma }}_n$ become larger than some $\gamma$. However, rare cases with $\mathrm{\Gamma }<\gamma$ may be present and actually dominate scattering events. We consider the statistical properties of these super-sharp resonances. We find that their number does not follow the fractal Weyl law conjectured for the bulk of the spectrum. We also test, for a simple model, the universal predictions of random matrix theory for density of states inside the gap and the hereby derived probability distribution of gap size.