Sharp upper bounds on resonances for perturbations of hyperbolic space

TL;DR

This paper derives explicit sharp upper bounds on the number of resonances for certain perturbations of hyperbolic space, with bounds depending only on geometric parameters, and confirms sharpness in specific scattering scenarios.

Contribution

It provides the first explicit, sharp upper bounds on resonance counts for perturbations of hyperbolic space, depending only on fundamental geometric quantities.

Findings

Established explicit upper bounds with a sharp constant

Bound depends only on dimension, perturbation radius, and volume

Confirmed sharpness of bounds in spherical obstacle scattering

Abstract

For certain compactly supported metric and/or potential perturbations of the Laplacian on $\mathbb{H}^{n+1}$, we establish an upper bound on the resonance counting function with an explicit constant that depends only on the dimension, the radius of the unperturbed region in $\mathbb{H}^{n+1}$, and the volume of the metric perturbation. This constant is shown to be sharp in the case of scattering by a spherical obstacle.