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arXiv · 2506.03577

Cantor spectrum for multidimensional quasi-periodic Schr\"odinger operators

Bernard Helffer·Qinghui Liu·Yanhui Qu·Qi Zhou
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Abstract

In this paper, we investigate the spectrum of a class of multidimensional quasi-periodic Schr\"odinger operators that exhibit a Cantor spectrum, which provides a resolution to a question posed by Damanik, Fillman, and Gorodetski \cite{DFG}. Additionally, we prove that for a dense set of irrational frequencies with positive Hausdorff dimension, the Hausdorff (and upper box) dimension of the spectrum of the critical almost Mathieu operator is positive, yet can be made arbitrarily small.

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Bernard Helffer, Qinghui Liu, Yanhui Qu, Qi Zhou. 2025-06-04. Cantor spectrum for multidimensional quasi-periodic Schr\"odinger operators. https://doi.org/10.1017/fms.2026.10242

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