Absolutely Continuous Spectrum of Perturbed Stark Operators

In this paper, we study the stability of the absolutely continuous spectrum of onedimensional Stark operators under various classes of perturbations. Stark Schrödinger operators describe behavior of the charged particle in the constant electric field. The absolutely continuous spectrum is a manifestation of the fact that the particle described by the operator propagates to infinity at a rather fast rate (see, e.g. [1]). It is therefore interesting to describe the classes of perturbations which preserve the absolutely continuous spectrum of the Stark operators. In the first part of this work, we study perturbations of Stark operators by decaying potetnials. This part is inspired by the recent work of Naboko and Pushnitski [12]. The general picture that we prove is very similar to the case of perturbations of free Schrödinger operators [8]. In accordance with physical intuition, however, the absolutely continuous spectrum is stable under stronger perturbations than in the free case. If in the free case the short range potentials preserving purely absolutely continuous spectrum of the free operator are given by condition (on the power scale) |q(x)| ≤ C(1 + |x|), in the Stark operator case the corresponding condition reads |q(x)| ≤ C(1 + |x|) 1 2 . If ǫ is allowed to be zero in the above bounds, imbedded eigenvalues may occur in both cases (see, e.g. [12], [13]). Moreover, in both cases if we allow potential to decay slower by an arbitrary function growing to infinity, very rich singular spectrum, such as a dense set of eigenvalues, may occur (see [11] for the free case and [12] for the Stark case for precise formulation and proofs of these results). The first part of this work draws the paprallel further, showing that the absolutely continuous spectrum of Stark operators is preserved under perturbations satisfying |q(x)| ≤ C(1 + |x|) 1 3 , in particular even in the regimes where a dense set of eignevalues occurs; hence in such cases these eigenvalues are genuinely imbedded. Similar results for the free case