APRG Seminar: 2025-03-05 - Jan Frahm

Speaker: Jan Frahm (Aarhus University, Denmark)

Title: Resonances for pseudo-Riemannian hyperbolic spaces

Abstract:
Pseudo-Riemannian hyperbolic spaces are generalizations of Riemannian hyperbolic spaces, including for instance de Sitter and anti-de Sitter spaces. They can be viewed as the pseudo-Riemannian analogs of Riemannian symmetric spaces of rank one. We show that for these spaces, the resolvent of the Laplace-Beltrami operator can be extended meromorphically across its spectrum to a double cover of the pointed complex plane. The poles of this extension are called (quantum) resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator forms a representation of the isometry group of the space, which we also describe concretely. This generalizes analogous results by Miatello-Will and Hilgert-Pasquale to the pseudo-Riemannian setting.