Absence of Absolutely Continuous Spectrum for Generic Quasi-Periodic Schr\"odinger Operators on the Real Line

David Damanik; Daniel Lenz

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arXiv · 1909.00378

Absence of Absolutely Continuous Spectrum for Generic Quasi-Periodic Schr\"odinger Operators on the Real Line

David Damanik·Daniel Lenz

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Abstract

We show that a generic quasi-periodic Schr\"odinger operator in $L^2(\mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schr\"odinger operator with the resulting potential has empty absolutely continuous spectrum.

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BibTeX RIS

David Damanik, Daniel Lenz. 2019-09-01. Absence of Absolutely Continuous Spectrum for Generic Quasi-Periodic Schr\"odinger Operators on the Real Line. https://arxiv.org/abs/1909.00378

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