Resonance asymptotics for Schrodinger operators on hyperbolic space

TL;DR

This paper investigates the distribution of resonances for Schrödinger operators on hyperbolic space, establishing bounds and asymptotics that depend on the potential's support and symmetry properties.

Contribution

It provides the first sharp resonance counting bounds in hyperbolic space and proves a Weyl law for radial potentials, extending resonance asymptotics to generic cases.

Findings

Established an upper bound for resonance counting function based on dimension and support.

Proved a Weyl law for radial potentials with finite support.

Derived resonance asymptotics in generic settings, including sector counts.

Abstract

We study the asymptotic distribution of resonances for scattering by compactly supported potentials in hyperbolic space. We first establish an upper bound for the resonance counting function that depends only on the dimension and the support of the potential. We then establish the sharpness of this estimate by proving the a Weyl law for the resonance counting function holds in the case of radial potentials vanishing to some finite order at the edge of the support. As an application of the existence of potentials that saturate the upper bound, we derive additional resonance asymptotics that hold in a suitable generic sense. These generic results include asymptotics for the resonance count in sectors.