Potential scattering and the continuity of phase-shifts

Jesse Gell-Redman; Andrew Hassell

paper

Potential scattering and the continuity of phase-shifts

arXiv:1112.3413

Abstract

Let $S (k)$ be the scattering matrix for a Schrödinger 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 $e^{2 i δ (k)}$ of S(k) $co n v er g in g t o 1 a s$ k \downarrow k_0 $. T hi ss h o w s t ha tt h ee i g e n v a l u eso f t h esc a tt er in g ma t r i x, a s a f u n c t i o n o f$ k $, d o n o t n ecess a r i l y ha v eco n t in u o u se x t e n s i o n s t oor a cr oss t h e v a l u e 1. I n p a r t i c u l a r t hi ss h o w s t ha t a ‘ mi cr o - L e v in so n t h eor e m^{'} f or n o n - ce n t r a l p o t e n t ia l s in$ \RR^3$ claimed in a 1989 paper of R. Newton is incorrect.

10 pages, 2 figures. Corrections made following suggestions of referee. Acknowledgements added