Angela Pasquale (Université de Lorraine) will speak in the geometry seminar on Monday 1st of April, at 2pm in the Salle de Profs. Angela’s title is Resonances of the Laplacian on products of rank-one symmetric spaces and her abstract is below.
Let $\Delta$ be the Laplacian on a Riemannian symmetric space $X=G/K$ of the noncompact type, and let $\sigma(\Delta)\subseteq \mathbb{C}$ denote its spectrum. The resolvent $(\Delta-z)^{-1}$ of $\Delta$ is a bounded operator on $L^2(X)$ which depends holomorphically on the complex variable $z\in \mathbb{C} \setminus \sigma(\Delta)$. Let $C^\infty(X)$ and $C_c^\infty(X)$ respectively denote the space of smooth and compactly-supported smooth functions on $X$. If we view the resolvent as a function on $C_c^\infty(X)$, then it has values in $C^\infty(X)$ and it often admits a meromorphic continuation across the spectrum $\sigma(\Delta)$, to a suitable Riemann surface above $\mathbb{C}$. The poles of the meromorphically extended resolvent are called resonances. The image of the residue operator at a resonance is a spherical $G$-module. The main problems are the existence and the localization of the resonances as well as the study of the corresponding $G$-modules.
Using the Helgason-Fourier analysis, one obtains an explicit formula for the resolvent as a singular intergral, and the study of the resonances of $\Delta$ reduces to the
meromorphic continuation of this integral. In this talk we look at low rank situations in which the space $X$ is a product of rank-one symmetric spaces.This talk is based on ongoing works with Joachim Hilgert (Universit\”at Paderborn) and Tomasz Przebinda (University of Oklahoma).