Sharp geometric upper bounds on resonances for surfaces with hyperbolic ends

TL;DR

This paper derives a precise geometric upper bound on the number of resonances for surfaces with hyperbolic ends, depending only on core volume and end parameters, and confirms its sharpness in key cases.

Contribution

It introduces a sharp, geometry-dependent upper bound on resonance counts for hyperbolic-ended surfaces, extending previous results to more general metrics and end types.

Findings

The upper bound depends only on core volume and end parameters.

The estimate is sharp for finite-area surfaces with cusp ends.

The bound matches known asymptotics for funnel ends with Dirichlet conditions.

Abstract

We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the core and the length parameters associated to the funnel or hyperbolic planar ends. Our estimate is sharp in that it reproduces the exact asymptotic constant in the case of finite-area surfaces with hyperbolic cusp ends, and also in the case of funnel ends with Dirichlet boundary condtiions.