Gluing semiclassical resolvent estimates via propagation of singularities

TL;DR

This paper introduces a method using semiclassical propagation of singularities to combine resolvent estimates, with applications to Schrödinger operators on asymptotically hyperbolic manifolds, leading to decay and smoothing results.

Contribution

It develops a general technique for gluing resolvent estimates via propagation of singularities, applicable to complex geometric settings with trapping.

Findings

Proved resolvent estimates for Schrödinger operators on asymptotically hyperbolic manifolds.

Established local exponential decay for wave propagators.

Demonstrated local smoothing effects for Schrödinger propagators.

Abstract

We use semiclassical propagation of singularities to give a general method for gluing together resolvent estimates. As an application we prove estimates for the analytic continuation of the resolvent of a Schr\"odinger operator for certain asymptotically hyperbolic manifolds in the presence of trapping which is sufficiently mild in one of several senses. As a corollary we obtain local exponential decay for the wave propagator and local smoothing for the Schr\"odinger propagator.