Resonances and Scattering Poles in Symmetric Spaces of Rank One

2017/03/08 by Hansen, Sönke, Hilgert, Joachim, Parthasarathy, Aprameyan · 1 citation
#FOS: Mathematics #Representation Theory (math.RT) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1703.02916

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.

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