Potential scattering and the continuity of phase-shifts

Mathematics – Spectral Theory

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

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

Scientific paper

Let $S(k)$ be the scattering matrix for a Schr\"odinger 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 $e^{2i\delta(k)}$ of S(k)$ converging to 1 as $k \downarrow k_0$. 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 $\RR^3$ claimed in a 1989 paper of R. Newton is incorrect.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Potential scattering and the continuity of phase-shifts does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Potential scattering and the continuity of phase-shifts, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Potential scattering and the continuity of phase-shifts will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-561848

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.