 01381 M. Christ and A. Kiselev
 Absolutely continuous spectrum of Stark operators
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Oct 16, 01

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Abstract. We prove several new results on the absolutely continuous spectra of
perturbed onedimensional
Stark operators. First, we find new classes of perturbations,
characterized mainly by smoothness
conditions, which preserve purely absolutely continuous spectrum. Then
we establish stability of
the absolutely continuous spectrum in more general situations, where
imbedded singular spectrum
may occur. We present two kinds of optimal conditions for the stability
of absolutely continuous
spectrum: decay and smoothness. In the decay direction, we show that a
sufficient (in the power scale) condition is
$q(x) \leq C(1+x)^{\frac{1}{4}\epsilon};$ in the smoothness
direction, a sufficient condition in H\"older classes is
$q \in C^{\frac{1}{2}+\epsilon}(\reals)$.
On the other hand, we show that there exist potentials which both
satisfy $q(x) \leq C(1+x)^{\frac14}$
and belong to $C^{\frac12}(\reals)$ for
which the spectrum becomes purely singular
on the whole real axis, so that the above results are optimal within
the scales considered.
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