. SP ] 1 7 O ct 2 00 1 ABSOLUTELY CONTINUOUS SPECTRUM OF STARK OPERATORS
نویسنده
چکیده
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; in the smoothness direction, a sufficient condition in Hölder classes is q ∈ C 12+ǫ(R). On the other hand, we show that there exist potentials which both satisfy |q(x)| ≤ C(1 + |x|)− 14 and belong to C 1
منابع مشابه
Preservation of the Absolutely Continuous Spectrum of Schrödinger Equation under Perturbations by Slowly Decreasing Potentials and A.e. Convergence of Integral Operators
X iv :m at h/ 96 10 21 6v 1 [ m at h. SP ] 1 O ct 1 99 6 Abstract. We prove new criteria of stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators under slowly decaying perturbations. As applications, we show that the absolutely continuous spectrum of the free and periodic Schrödinger operators is preserved under perturbations by all potentials V (x) satisfying...
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