Upper and lower bounds on resonances for manifolds hyperbolic near infinity

TL;DR

This paper establishes sharp upper and lower bounds on the number of resonances for hyperbolic-like manifolds, correcting previous errors and providing a Poisson formula linking wave traces and scattering resonances.

Contribution

It completes the proof of the optimal resonance counting bounds and introduces a Poisson formula for manifolds with hyperbolic infinity, advancing understanding of scattering theory.

Findings

Proves the $O(r^{n+1})$ upper bound on resonance counting functions.

Establishes an $r^{n+1}$ lower bound for scattering poles.

Corrects a mistake in the existing literature regarding resonance bounds.

Abstract

For a conformally compact manifold that is hyperbolic near infinity and of dimension $n+1$, we complete the proof of the optimal $O(r^{n+1})$ upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an $r^{n+1}$ lower bound on the counting function for scattering poles.