Perturbations of one-dimensional Schrodinger operators preserving the absolutely continuous spectrum

Citation

Killip, Rowan (2001) Perturbations of one-dimensional Schrodinger operators preserving the absolutely continuous spectrum. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-09062005-102553

Abstract

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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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