Resonances for asymptotically hyperbolic manifolds: Vasy's method revisited

TL;DR

This paper revisits Vasy's method for establishing the meromorphic continuation of the resolvent on asymptotically hyperbolic manifolds, simplifying the approach using standard pseudodifferential techniques.

Contribution

It provides a simplified, effective definition of resonances in asymptotically hyperbolic manifolds by identifying them with poles of inverses of Fredholm differential operators.

Findings

Simplified Vasy's method using standard pseudodifferential techniques.

Established more natural invertibility properties of the Fredholm family.

Extended the effective definition of resonances to hyperbolic settings.

Abstract

We revisit Vasy's method for showing meromorphy of the resolvent for (even) asymptotically hyperbolic manifolds. It provides an effective definition of resonances in that setting by identifying them with poles of inverses of a family of Fredholm differential operators. In the Euclidean case the method of complex scaling made this available since the 70's but in the hyperbolic case an effective definition was not known until recently. Here we present a simplified version which relies only on standard pseudodifferential techniques and estimates for hyperbolic operators. As a byproduct we obtain more natural invertibility properties of the Fredholm family.