Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Potential scattering and the continuity of phase-shifts   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.

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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
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The categories of this paper are: alg-geom
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The categories of this paper are: acc-phys
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The categories of this paper are: q-fin
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The categories of this paper are: eess
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin