Arkiv för Matematik

Absolutely continuous spectrum of Stark operators

Michael Christ and Alexander Kiselev

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Abstract

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 qC1/2+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.

Note

The first author was supported in part by NSF grant DMS-9970660 and completed this research while on appointment as a Miller Research Professor in the Miller Institute for Basic Research in Science.

Note

The second author was supported in part by NSF grant DMS-9801530 and by an Alfred P. Sloan Fellowship.

Article information

Source
Ark. Mat., Volume 41, Number 1 (2003), 1-33.

Dates
Received: 5 November 2001
Revised: 29 April 2002
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898789

Digital Object Identifier
doi:10.1007/BF02384565

Mathematical Reviews number (MathSciNet)
MR1971938

Zentralblatt MATH identifier
1046.34095

Rights
2003 © Institut Mittag-Leffler

Citation

Christ, Michael; Kiselev, Alexander. Absolutely continuous spectrum of Stark operators. Ark. Mat. 41 (2003), no. 1, 1--33. doi:10.1007/BF02384565. https://projecteuclid.org/euclid.afm/1485898789


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