Super-sharp resonances in chaotic wave scattering

TL;DR

This paper investigates the statistical properties of super-sharp resonances in chaotic wave scattering, revealing deviations from expected fractal Weyl law behavior and testing universal predictions of random matrix theory.

Contribution

It demonstrates that super-sharp resonances do not follow the fractal Weyl law and tests the applicability of random matrix theory predictions to their density and gap distribution.

Findings

Super-sharp resonances are fewer than predicted by the fractal Weyl law.

The density of states inside the gap aligns with random matrix theory.

The probability distribution of gap sizes matches theoretical predictions.

Abstract

Wave scattering in chaotic systems can be characterized by its spectrum of resonances, $z_n=E_n-i\frac{\Gamma_n}{2}$, where $E_n$ is related to the energy and $\Gamma_n$ 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 $\Gamma_n$ become larger than some $\gamma$. However, rare cases with $\Gamma<\gamma$ 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.