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On gaps in the spectra of quasiperiodic Schr\"odinger operators with discontinuous monotone potentials
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We show that, for one-dimensional discrete Schr\"odinger operators, stability of Anderson localization under a class of rank one perturbations implies absence of intervals in spectra. The argument is based on well-known result of Gordon and del Rio--Makarov--Simon, combined with a way to consider perturbations whose ranges are not necessarily cyclic. The main application of the results is showing that a class of quasiperiodic operators with sawtooth-like potentials, for which such a version of stable localization is known, has Cantor spectra. We also obtain several results on gap filling under rank one perturbations for some general (not necessarily monotone) classes of quasiperiodic operators with discontinuous potentials.
Forward citations
Cited by 2 Pith papers
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On a Question of Poltoratski
A sparse potential is built so that the essential spectrum is [-2,2] and the spectral measure of the boundary vector stays non-Rajchman for every rank-one perturbation.
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On Fractal Continuity Properties of Certain One-Dimensional Schr\"odinger Operators
The paper gives explicit constructions of half-line and whole-line Schrödinger operators with prescribed spectral fractal dimensions, but Theorem 1.1's proof has a false step.
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