Equidistribution of phase shifts in trapped scattering

Abstract

We prove an equidistribution result for the eigenvalues of the scattering matrix associated to an operator of the form $-h^2\Delta + V-1$, where $V\in C_c^\infty(\mathbb{R}^d)$ is a compactly supported potential, under the assumption that the incoming and outgoing sets of the classical dynamics have zero Liouville measure. This extends a recent result of Gell-Redman, Hassell and Zelditch, where the authors proved equidistribution of the eigenvalues of the scattering matrix under the assumption that the trapped set is empty.