Absolutely continuous spectrum of Stark operators
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
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- General Mathematics
- 4904 Pure mathematics
- 0101 Pure Mathematics
Citation
Published In
DOI
ISSN
Publication Date
Volume
Issue
Start / End Page
Related Subject Headings
- General Mathematics
- 4904 Pure mathematics
- 0101 Pure Mathematics