A Poisson Formula for the Wave Propagator on Schwarzschild-de Sitter Backgrounds

TL;DR

This paper develops a Poisson formula for the wave propagator on Schwarzschild-de Sitter backgrounds, linking wave behavior to scattering resonances, and extends previous formulas to non-compactly supported potentials.

Contribution

It introduces a Poisson formula applicable to non-compactly supported potentials, including those from Schwarzschild-de Sitter metrics, expanding the scope of previous results.

Findings

Established a Poisson formula for SdS wave propagators

Connected wave propagators with scattering resonances for non-compact potentials

Extended classical Poisson formulas to non-compact support scenarios

Abstract

This paper proposes a Poisson formula for the wave propagator of the Schwarzschild--de Sitter (SdS) metric. That is done by proving a Poisson formula relating wave propagators and scattering resonances for a class of non-compactly supported potentials on the real line. That class includes the Regge--Wheeler potentials obtained from separation of variables for SdS. The novelty lies in allowing non-compact supports -- all exact Poisson formulae of Lax--Phillips, Melrose, and other authors required compactness of the support of the perturbation.