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

Citation

Killip, Rowan Brett (2001) Perturbations of One-Dimensional Schrödinger Operators Preserving the Absolutely Continuous Spectrum. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/2t54-0b07. https://resolver.caltech.edu/CaltechETD:etd-09062005-102553

Abstract

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.

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