Equidistribution of phase shifts in semiclassical potential scattering

Consider the Hamiltonian H := h(2) Delta + V - E where Delta is the positive Laplacian on R-d, V is an element of C-0(infinity) (R-d) is a smooth, compactly supported potential, E > 0 is an energy level, and h > 0 is a semiclassical parameter. We study the eigenvalues of the scattering matrix S-h(E), which lie on the unit circle S-1 subset of C due to the unitarity of S-h(E). Under an appropriate hypothesis on the classical dynamical flow corresponding to H, we show that in the limit h -> 0, the eigenvalues are asymptotically equidistributed on the unit circle away from the point 1 is an element of S-1.