Absolutely Continuous Spectrum of Stark Operators
نویسندگان
چکیده
Abstract. We prove several new results on the absolutely continuous spectrum of perturbed one-dimensional Stark operators. First, we find new classes of potentials, characterized mainly by smoothness condition, 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 decay direction, we show that the sufficient (on the power scale) condition is |q(x)| ≤ C(1 + |x|)− 1 4; in smoothness direction the sufficient condition in Hölder classes is q ∈ C 12+ǫ(R). On the other hand, we show that there exist potentials which satisfy |q(x)| ≤ C(1 + |x|)− 1 4 and belong to C 1
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