Resonances for asymptotically hyperbolic manifolds: Vasy’s method revisited

Abstract

We revisit Vasy’s method ([27] and [28]) 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 till [27] and [28]. 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.