Equidistribution of phase shifts in semiclassical potential scattering

Abstract

Consider the Hamiltonian H:= h2 Δ +V-E where Δ is the positive Laplacian on Rd, V in C0(Rd) 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 Sh(E), which lie on the unit circle S1 C due to the unitarity of Sh(E). Under an appropriate hypothesis on the classical dynamical flow corresponding to H, we show that in the limit h to 0, the eigenvalues are asymptotically equidistributed on the unit circle away from the point 1 in S1.