Absolutely continuous spectrum of Stark operators

Journal Article (Journal Article)

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 Holder classes is q ∈ C1/2+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 C1/2(R) for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.

Full Text

Duke Authors

Cited Authors

  • Christ, M; Kiselev, A

Published Date

  • January 1, 2003

Published In

Volume / Issue

  • 41 / 1

Start / End Page

  • 1 - 33

International Standard Serial Number (ISSN)

  • 0004-2080

Digital Object Identifier (DOI)

  • 10.1007/BF02384565

Citation Source

  • Scopus