Potential Scattering and the Continuity of Phase-shifts

Let S(k) be the scattering matrix for a Schrödinger operator (Laplacian plus potential) on Rn 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 they stay away from 1. We give examples of smooth, compactly supported potentials on Rn for which (i) the scattering matrix S(k) does not have 1 as an eigenvalue for any k > 0, and (ii) there exists k0 > 0 such that there is an analytic eigenvalue branch e2iδ(k) of S(k) converging to 1 as k ↓ k0. 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 R3 claimed in a 1989 paper of R. Newton is incorrect.