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 $S_h$, at some fixed energy $E$, for semiclassical Schrödinger operators on $\mathbb{R}^d$ that are perturbations of the free Hamiltonian $h^2 \Delta $ on $L^2(\mathbb{R}^d)$ by a potential $V$ with polynomial decay. Our assumption is that $V(x) \sim |x|^{-\alpha } v(\hat x)$ as $x \to \infty $, $\hat x = x/|x|$, for some $\alpha> d$, with corresponding derivative estimates. In the semiclassical limit $h \to 0$, we show that the atomic measure on the unit circle defined by these eigenvalues, after suitable scaling in $h$, tends to a measure $\mu $ on $\mathbb{S}^1$. Moreover, $\mu $ is the pushforward from $\mathbb{R}$ to $\mathbb{R} / 2 \pi \mathbb{Z} = \mathbb{S}^1$ of a homogeneous distribution. As a corollary we obtain an asymptotic formula for the accumulation of phase shifts in a sector of $\mathbb{S}^1$. 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.