Absolutely continuous spectrum of Stark operators

Michael Christ Department of Mathematics, University of California Alexander Kiselev Department of Mathematics, University of Chicago

TBD mathscidoc:1701.332991

Arkiv for Matematik, 41, (1), 1-33, 2001.11
We prove several new results on the absolutely continuous spectra of perturbed one-dimensional 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 |$q$($x$)|≤$C$(1+|$x$|)^{−1/4−ε}; in the smoothness direction, a sufficient condition in Hölder classes is$q$∈$C$^{1/2+ε}($R$). On the other hand, we show that there exist potentials which both satisfy |$q$($x$)|≤$C$(1+|$x$|)^{−1/4}and belong to$C$^{1/2}($R$) 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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@inproceedings{michael2001absolutely,
  title={Absolutely continuous spectrum of Stark operators},
  author={Michael Christ, and Alexander Kiselev},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203605655928800},
  booktitle={Arkiv for Matematik},
  volume={41},
  number={1},
  pages={1-33},
  year={2001},
}
Michael Christ, and Alexander Kiselev. Absolutely continuous spectrum of Stark operators. 2001. Vol. 41. In Arkiv for Matematik. pp.1-33. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203605655928800.
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