Analytic continuation and semiclassical resolvent estimates on asymptotically hyperbolic spaces

TL;DR

This paper develops a parametrix for high-energy analytic continuation of the resolvent on asymptotically hyperbolic spaces, leading to improved resolvent estimates in non-trapping scenarios.

Contribution

It introduces a novel parametrix construction for the resolvent's analytic continuation on perturbed hyperbolic spaces, advancing high-energy resolvent estimates.

Findings

Established non-trapping high-energy resolvent estimates

Constructed a parametrix for the analytic continuation of the resolvent

Extended results to small perturbations of hyperbolic space

Abstract

In this paper we construct a parametrix for the high-energy asymptotics of the analytic continuation of the resolvent on a Riemannian manifold which is a small perturbation of the Poincar\'e metric on hyperbolic space. As a result, we obtain non-trapping high energy estimates for this analytic continuation.