Approximation and Equidistribution of Phase Shifts: Spherical Symmetry

Abstract

Consider a semiclassical Hamiltonian HV,h := h2 Δ + V − E, where h > 0 is a semiclassical parameter, Δ is the positive Laplacian on Rd , V is a smooth, compactly supported central potential function and E > 0 is an energy level. In this setting the scattering matrix Sh(E) is a unitary operator on L2(Sd−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 Sh(E) can be divided into two classes: a finite number ∼ cd (R √ E/h)d−1, as h → 0, where B(0, R) is the convex hull of the support of the potential, that equidistribute around the unit circle, and the remainder ..