# Resonances and Scattering Poles in Symmetric Spaces of Rank One

## TL;DR

This paper explores the relationship between resolvent and scattering kernels on rank one symmetric spaces, establishing pole correspondence and providing new proofs for Helgason's conjecture.

## Contribution

It introduces a novel boundary value approach to relate resolvent and scattering poles, and offers a new proof of Helgason's conjecture for rank one symmetric spaces.

## Key findings

- Poles of resolvent and scattering kernels agree in a half-plane.
- Residues of resolvent and scattering kernels correspond under boundary values.
- New proof of Helgason's conjecture for rank one symmetric spaces.

## Abstract

We relate resolvent and scattering kernels for the Laplace operator on Riemannian symmetric spaces of rank one via boundary values in the sense of Kashiwara-Oshima. From this, we derive that the poles of the corresponding meromorphic continuations agree in a half-plane, and the residues correspond to each other under the boundary value map, so in particular the multiplicities agree as well. In the opposite half-plane, which is the square root of the resolvent set, the resolvent has no poles, whereas the scattering poles agree with the poles of the standard Knapp--Stein intertwiner. As a by-product of the underlying ideas, we obtain a new and self-contained proof of Helgason's conjecture for distributions in the case of rank one symmetric spaces.