The distribution of phase shifts for semiclassical potentials with polynomial decay

TL;DR

This paper studies the asymptotic distribution of phase shifts for semiclassical Schrödinger operators with polynomially decaying potentials, revealing a limiting measure on the unit circle related to the potential's decay rate.

Contribution

It extends previous analyses to potentials with polynomial decay, establishing the limiting distribution of eigenvalues of the scattering matrix in the semiclassical limit.

Findings

The eigenvalues' distribution converges to a measure on the unit circle as h → 0.

The limiting measure is a pushforward of a homogeneous distribution depending on decay rate.

An asymptotic formula for phase shift accumulation in sectors of the circle is derived.

Abstract

This is the third paper in a series 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, $S_h(E)$, for semiclassical Schr\"odinger operators on $\mathbb{R}^d$ which are perturbations of the free Hamiltonian by a potential $V$ with polynomial decay. Our assumption is that $V(x) \sim |x|^{-\alpha} v(\hat x)$ as $x \to \infty$, 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 $\nu$ of order $\beta$ depending on the dimension $d$ and the rate of decay $\alpha$ of the potential function. 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 of the second author and Wunsch on the classical Hamiltonian dynamics and semiclassical Poisson operator to the class of potentials under consideration here.