Resonance asymptotics for Schrödinger operators on hyperbolic space

Abstract

We study the asymptotic distribution of resonances for scattering by compactly supported potentials in $H^{n+1}$. 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.