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Absolutely continuous spectrum of Stark operators

Publication ,  Journal Article
Christ, M; Kiselev, A
Published in: Arkiv for Matematik
January 1, 2003

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.

Duke Scholars

Published In

Arkiv for Matematik

DOI

ISSN

0004-2080

Publication Date

January 1, 2003

Volume

41

Issue

1

Start / End Page

1 / 33

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics
 

Citation

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Christ, M., & Kiselev, A. (2003). Absolutely continuous spectrum of Stark operators. Arkiv for Matematik, 41(1), 1–33. https://doi.org/10.1007/BF02384565
Christ, M., and A. Kiselev. “Absolutely continuous spectrum of Stark operators.” Arkiv for Matematik 41, no. 1 (January 1, 2003): 1–33. https://doi.org/10.1007/BF02384565.
Christ M, Kiselev A. Absolutely continuous spectrum of Stark operators. Arkiv for Matematik. 2003 Jan 1;41(1):1–33.
Christ, M., and A. Kiselev. “Absolutely continuous spectrum of Stark operators.” Arkiv for Matematik, vol. 41, no. 1, Jan. 2003, pp. 1–33. Scopus, doi:10.1007/BF02384565.
Christ M, Kiselev A. Absolutely continuous spectrum of Stark operators. Arkiv for Matematik. 2003 Jan 1;41(1):1–33.
Journal cover image

Published In

Arkiv for Matematik

DOI

ISSN

0004-2080

Publication Date

January 1, 2003

Volume

41

Issue

1

Start / End Page

1 / 33

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics