The distribution of phase shifts for semiclassical potentials with polynomial decay

This is the 3rd paper in a series [6, 9] analyzing the asymptotic distribution of the phase shifts in the semiclassical limit. We analyze the distribution of phase shifts, or equivalently, eigenvalues of the scattering matrix Sh, at some fixed energy E, for semiclassical Schrödinger operators on Rd that are perturbations of the free Hamiltonian h² on L²(Rd) by a potential V with polynomial decay. Our assumption is that V(x) ~ |x|-av(ˆx ) as x?8, x = x/|x|, for somea > d, with corresponding derivative estimates. In the semiclassical limit h ? 0, we show that the atomic measure on the unit circle defined by these eigenvalues, after suitable scaling in h, tends to a measure μ on S1. Moreover, μ is the pushforward from R to R/2pZ = S1 of a homogeneous distribution. As a corollary we obtain an asymptotic formula for the accumulation of phase shifts in a sector of S1. The proof relies on an extension of results in [14] on the classical Hamiltonian dynamics and semiclassical Poisson operator to the larger class of potentials under consideration here.