| We study the stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators with periodic potentials. Specifically, it is proved that any perturbation of the potential which is square integrable preserves the essential support (and multiplicity) of the absolutely continuous spectrum. This is optimal in terms of Lebesgue spaces and, for zero background potential, it answers in the affirmative a conjecture of Kiselev, Last and Simon.; By adding constraints on the Fourier transform of the perturbation, it is possible to relax the decay assumptions. If q is identically zero, a similar result is proved which is local in energy. |
