Potential scattering and the continuity of phase-shifts

TL;DR

This paper constructs examples demonstrating that eigenvalues of the scattering matrix for Schrödinger operators can behave discontinuously around the value 1, challenging previous assumptions in scattering theory.

Contribution

It provides explicit examples showing eigenvalues of the scattering matrix can lack continuous extension across 1, refuting a prior theorem by Newton.

Findings

Eigenvalues of S(k) can approach 1 without eigenvalue crossing.

Examples show S(k) can lack eigenvalue at 1 for all k.

Discontinuous eigenvalue behavior invalidates a previous Levinson theorem.

Abstract

Let $S(k)$ be the scattering matrix for a Schr\"odinger operator (Laplacian plus potential) on $\RR^n$ with compactly supported smooth potential. It is well known that $S(k)$ is unitary and that the spectrum of $S(k)$ accumulates on the unit circle only at 1; moreover, $S(k)$ depends analytically on $k$ and therefore its eigenvalues depend analytically on $k$ provided the values stay away from 1. We give examples of smooth, compactly supported potentials on $\RR^n$ for which (i) the scattering matrix $S(k)$ does not have 1 as an eigenvalue for any $k > 0$, and (ii) there exists $k_0 > 0$ such that there is an analytic eigenvalue branch $e^{2i\delta(k)}$ of S(k)$ converging to 1 as $k \downarrow k_0$. This shows that the eigenvalues of the scattering matrix, as a function of $k$, do not necessarily have continuous extensions to or across the value 1. In particular this shows that a `micro-Levinson theorem' for non-central potentials in $\RR^3$ claimed in a 1989 paper of R. Newton is incorrect.