Approximation and equidistribution of phase shifts: spherical symmetry

Consider a semiclassical Hamiltonian \begin{equation*} H_{V, h} := h^{2} \Delta + V - E \end{equation*} where $h>0$ is a semiclassical parameter, $\mathrm{\Delta }$ is the positive Laplacian on ${\mathbb{R}}^d$, $V$ is a smooth, compactly supported central potential function and $E>0$ is an energy level. In this setting the scattering matrix $S_h(E)$ is a unitary operator on $L^2({\mathbb{S}}^{d-1})$, hence with spectrum lying on the unit circle; moreover, the spectrum is discrete except at $1$. We show under certain additional assumptions on the potential that the eigenvalues of $S_h(E)$ can be divided into two classes: a finite number $\sim c_d (R\sqrt{E}/h)^{d-1} $,as$h \to 0$,where$B(0, R)$ is the convex hull of the support of the potential, that equidistribute around the unit circle, and the remainder that are all very close to $1$. Semiclassically, these are related to the rays that meet the support of, and hence are scattered by, the potential, and those that do not meet the support of the potential, respectively. A similar property is shown for the obstacle problem in the case that the obstacle is the ball of radius $R$.