Analytic continuation and semiclassical resolvent estimates on asymptotically hyperbolic spaces

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.