Abstract:
We consider decaying oscillatory perturbations of periodic Schr\"odinger
operators on the half line. More precisely, the perturbations we study
satisfy a generalized bounded variation condition at infinity and an $L^p$
decay condition. We show that the absolutely continuous spectrum is
preserved, and give bounds on the Hausdorff dimension of the singular part
of the resulting perturbed measure. Under additional assumptions, we
instead show that the singular part embedded in the essential spectrum is
contained in an explicit countable set. Finally, we demonstrate that this
explicit countable set is optimal. That is, for every point in this set
there is an open and dense class of periodic Schr\"odinger operators for
which an appropriate perturbation will result in the spectrum having an
embedded eigenvalue at that point.
