Distribution of Resonances for Hyperbolic Surfaces

Abstract

We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by counting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspects of the resonance distribution that have attracted attention recently: the fractal Weyl law, the spectral gap, and the concentration of decay rates.

2000 AMS Subject Classification:

• 58 J50

Keywords:

• spectral theory
• resonances
• hyperbolic surfaces

Acknowledgments

I would like to thank Maciej Zworski for encouragement to develop this project, and for helpful comments and suggestions along the way.

FUNDING

This research was supported in part by NSF grant DMS-0901937.