On the absolutely continuous spectrum of Stark operators (Q1400912)

The invariance of the absolutely continuous spectrum of the perturbed one-dimensional Stark operator is studied. The method used allows one to treat potentials which have a singular part which is uniformly in \(L^{1}_{loc}\) and a smooth part which grows at infinity at a rate arbitrarily close to that of the constant electric field. It is proved that for a rather large class of such potentials, the absolutely continuous spectrum of the perturbed Stark operator coincides with \(\mathbb{R}\). The general result can be applied to periodic perturbations which are in \(L^{1}(\mathbb{T})\cap{H^{-1/2}(\mathbb{T})}\) and to potentials considered by \textit{M. Christ} and \textit{A. Kiselev} [J. Am. Math. Soc., 11, 771-797 (1998; Zbl 0899.34051)].