Resonances of the Laplacian on Riemannian symmetric spaces of the noncompact type of rank 2

Abstract: Let $X=G/K$ be a Riemannian symmetric space of non-compact type and let $\Delta$ be the positive Laplacian of $X$, with spectrum $\sigma(\Delta)$. Then the resolvent $R(z)=(\Delta-z)^{-1}$ is a holomorphic function on $\mathbb{C}\setminus \sigma(\Delta)$ with values in the space of bounded linear operators on $L^2(X)$. If $R$ admits a meromorphic continuation across $\sigma(\Delta)$, then the poles of the meromorphically extended resolvent are called the resonances of $\Delta$. At present, there are no general results on the existence and the nature of resonances on a general $X=G/K$. In this talk, we will mostly focus on the case of rank two.

This is part of a joint project with J. Hilgert (Paderborn University) and T. Przebinda (University of Oklahoma).

analysis of PDEsnumber theoryrepresentation theory

Audience: researchers in the topic

Harmonic Analysis and Symmetric Spaces 2021