Potential scattering and the continuity of phase-shifts

Abstract

Let S(k) be the scattering matrix for a Schrödinger 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 2iδ(k) of S(k) converging to 1 as k ↓ 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.

Locations

• Mathematical Research Letters - View - PDF
• arXiv (Cornell University) - View - PDF