A note on the resonance counting function for surfaces with cusps

TL;DR

This paper establishes precise upper bounds on the number of resonances for the Laplacian on finite volume hyperbolic cusp surfaces, providing a Weyl law with explicit remainder estimates at high frequencies.

Contribution

It introduces sharp upper bounds for resonance counts and derives a Weyl asymptotic with a specific remainder term for surfaces with cusps.

Findings

Proves sharp upper bounds for resonance counts in frequency boxes.

Derives a Weyl law for resonance distribution with an $O(T^{3/2})$ remainder.

Enhances understanding of spectral properties of hyperbolic cusp surfaces.

Abstract

We prove sharp upper bounds for the number of resonances in boxes of size 1 at high frequency for the Laplacian on finite volume surfaces with hyperbolic cusps. As a corollary, we obtain a Weyl asymptotic for the number of resonances in balls of size $T \to \infty$ with remainder $O(T^{3/2})$.