Equidistribution of phase shifts in semiclassical potential scattering

Consider a semiclassical Hamiltonian $H := h^{2} \Delta + V - E$ where $\Delta$ is the positive Laplacian on $\mathbb{R}^{d}$, $V \in C^{\infty}_{0}(\mathbb{R}^{d})$ and $E > 0$ is an energy level. We prove that under an appropriate dynamical hypothesis on the Hamilton flow corresponding to $H$, the eigenvalues of the scattering matrix $S_{h}(V)$ define a measure on $\mathbb{S}^{1}$ that converges to Lebesgue measure away from $1 \in \mathbb{S}^{1}$ as $h \to 0$.