Analytic continuation and high energy estimates for the resolvent of the Laplacian on forms on asymptotically hyperbolic spaces

TL;DR

This paper establishes the analytic continuation and high energy estimates for the resolvent of the Laplacian on differential forms in asymptotically hyperbolic spaces, extending scalar techniques to non-scalar operators via a Minkowski space analogy.

Contribution

It introduces a novel approach to handle non-scalar Laplacians on forms by relating them to Minkowski space problems, enabling analytic continuation and high energy estimates.

Findings

Achieved analytic continuation of the resolvent on forms.

Derived high energy estimates in strips for the resolvent.

Extended scalar problem techniques to non-scalar operators.

Abstract

We show the analytic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces on differential forms, including high energy estimates in strips. This is achieved by placing the spectral family of the Laplacian within the framework developed, and applied to scalar problems, by the author recently, roughly by extending the problem across the boundary of the compactification of the asymptotically hyperbolic space in a suitable manner. The main novelty is that the non-scalar nature of the operator is dealt with by relating it to a problem on an asymptotically Minkowski space to motivate the choice of the extension across the conformal boundary.