Approximation and equidistribution of phase shifts: spherical symmetry

TL;DR

This paper studies the spectral distribution of the scattering matrix for semiclassical Hamiltonians with spherical symmetry, showing that eigenvalues split into equidistributed phases and those close to 1, related to classical scattering rays.

Contribution

It demonstrates that under certain conditions, the eigenvalues of the scattering matrix split into two classes with distinct asymptotic behaviors, revealing a semiclassical phase distribution pattern.

Findings

Eigenvalues split into equidistributed phases and near 1.

Number of equidistributed eigenvalues scales with (R√E/h)^{d-1}.

Similar results apply to the obstacle problem with a spherical obstacle.

Abstract

Consider a semiclassical Hamiltonian \begin{equation*} H_{V, h} := h^{2} \Delta + V - E \end{equation*} where $h > 0$ is a semiclassical parameter, $\Delta$ is the positive Laplacian on $\mathbb{R}^{d}$, $V$ is a smooth, compactly supported central potential function and $E > 0$ is an energy level. In this setting the scattering matrix $S_h(E)$ is a unitary operator on $L^2(\mathbb{S}^{d-1})$, 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 $S_h(E)$ can be divided into two classes: a finite number $\sim c_d (R\sqrt{E}/h)^{d-1} $, as $h \to 0$, where $B(0, R)$ is the convex hull of the support of the potential, that equidistribute around the unit circle, and the remainder that are all very close to $1$. Semiclassically, these are related to the rays that meet the support of, and hence are scattered by, the potential, and those that do not meet the support of the potential, respectively. A similar property is shown for the obstacle problem in the case that the obstacle is the ball of radius $R$.