The Distribution of Phase Shifts for Semiclassical Potentials with Polynomial Decay

Abstract

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 ShSh⁠, at some fixed energy EE⁠, for semiclassical Schrödinger operators on RdRd that are perturbations of the free Hamiltonian h2Δh2Δ on L2(Rd)L2(Rd) by a potential VV with polynomial decay. Our assumption is that V(x)∼|x|−αv(x^)V(x)∼|x|−αv(x^) as x→∞x→∞⁠, x^=x/|x|x^=x/|x|⁠, for some α>dα>d⁠, with corresponding derivative estimates. In the semiclassical limit h→0h→0⁠, we show that the atomic measure on the unit circle defined by these eigenvalues, after suitable scalin..