The High Energy Distribution of Scattering Phase Shifts of Schrödinger Operators in Hyperbolic Space

We prove a trace formula for the high energy limit of the scattering phase shifts of Schrödinger operators with short range real valued potentials in hyperbolic space; it relates the scattering shifts and the geodesic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>X</mml:mi> </mml:math> -ray transform of the potential. This extends a result of Bulger and Pushnitski for Schrödinger operators in Euclidean space. As an application, we prove that the high energy limit of the phase shifts uniquely determines radial potentials which are monotone and decay super-exponentially. This extends a result of Levinson for potential perturbations of the Euclidean Laplacian to this special class of potentials in hyperbolic space.