Abstract：

We discuss existence and inverse results for Schr\"odinger operators on the hyperbolic space $\mathbb{H}^{n+1}$, with smooth, compactly supported potentials. Such results have been known in the Euclidean setting for several decades, based on the wave/heat trace expansions as well as asymptotics of the scatteringphase. We will present a full picture of the corresponding theory in the hyperbolic setting and highlight some of the key differences from the Euclidean case. This talk represents joint work with Yiran Wang.