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arXiv · 0906.3337

Spectral Properties of Limit-Periodic Schr\"odinger Operators

David Damanik·Zheng Gan
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Abstract

We investigate the spectral properties of Schr\"odinger operators in l^2(Z) with limit-periodic potentials. The perspective we take was recently proposed by Avila and is based on regarding such potentials as generated by continuous sampling along the orbits of a minimal translation of a Cantor group. This point of view allows one to separate the base dynamics and the sampling function. We show that for any such base dynamics, the spectrum is a Cantor set of positive Lebesgue measure and purely absolutely continuous for a dense set of sampling functions, and it is a Cantor set of zero Lebesgue measure and purely singular continuous for a dense G_\delta set of sampling functions.

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David Damanik, Zheng Gan. 2009-06-18. Spectral Properties of Limit-Periodic Schr\"odinger Operators. https://arxiv.org/abs/0906.3337

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