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arXiv · 1909.04429

Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum

Svetlana Jitomirskaya·Igor Krasovsky
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Abstract

We prove almost Lipshitz continuity of spectra of singular quasiperiodic Jacobi matrices and obtain a representation of the critical almost Mathieu family that has a singularity. This allows us to prove that the Hausdorff dimension of its spectrum is not larger than 1/2 for all irrational frequencies, solving a long-standing problem. Other corollaries include two very elementary proofs of zero measure of the spectrum (Problem 5 in [41]) and a similar Hausdorff dimension result for the quantum graph graphene.

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Svetlana Jitomirskaya, Igor Krasovsky. 2019-09-10. Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum. https://arxiv.org/abs/1909.04429

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