Potential scattering and the continuity of phase-shifts

Let $S(k)$ be the scattering matrix for a Schr"odinger operator (Laplacian plus potential) on ${\text{RR}}^n$ 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 ${\text{RR}}^n$ 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)$convergingto1as$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 ${\text{RR}}^3$ claimed in a 1989 paper of R. Newton is incorrect.