Perturbations of One-Dimensional Schrödinger Operators Preserving the Absolutely Continuous Spectrum

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. We study the stability of the absolutely continuous spectrum of one-dimensional Schrodinger operators [...] with periodic potentials q(x). Specifically, it is proved that any perturbation of the potential, [...], preserves the essential support (and multiplicity) of the absolutely continuous spectrum. This is optimal in terms of [...] spaces and, for [...], it answers in the affirmative a conjecture of Kiselev, Last and Simon. By adding constraints on the Fourier transform of V, it is possible to relax the decay assumptions on V. It is proved that if [...] and [...] is uniformly locally square integrable, then preservation of the a.c. spectrum still holds. If we assume that [...], still stronger results follow: if [...] and [...] is square integrable on an interval [...], then the interval [...] is contained in the essential support of the absolutely continuous spectrum of the perturbed operator