Mathematical Research Letters

Volume 19 (2012)

Number 3

Potential scattering and the continuity of phase-shifts

Pages: 719 – 729



Jesse Gell-Redman (Department of Mathematics, Australian National University, Canberra, Australia)

Andrew Hassell (Department of Mathematics, Australian National University, Canberra, Australia)


Let $S(k)$ be the scattering matrix for a Schrödinger operator (Laplacian plus potential) on $\RR^n$ with compactlysupported smooth potential. It is well known that $S(k)$ is unitary and that the spectrum of $S(k)$ accumulates on theunit circle only at $1$; moreover, $S(k)$ depends analytically on $k$ and therefore its eigenvalues dependanalytically on $k$ provided they stay away from $1$.

We give examples of smooth, compactly supported potentials on $\RR^n$ for which (i) the scattering matrix $S(k)$ doesnot have $1$ as an eigenvalue for any $k > 0$, and (ii) there exists $k_0 > 0$ such that there is an analyticeigenvalue branch ${\rm e}^{2{\rm i}\delta(k)}$ of $S(k)$ converging to $1$ as $k \downarrow k_0$. This shows that theeigenvalues of the scattering matrix, as a function of $k$, do not necessarily have continuous extensions to or acrossthe 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.

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