POTENTIAL SCATTERING AND THE CONTINUITY OF PHASE-SHIFTS

Let S(k) be the scattering matrix for a Schrodinger operator (Laplacian plus potential) on R-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 they stay away from 1. We give examples of smooth, compactly supported potentials on R-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 down arrow 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 R-3 claimed in a 1989 paper of R. Newton is incorrect.