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

2019/09/10 by Svetlana Jitomirskaya, Jitomirskaya, Svetlana, Igor Krasovsky +1 · 1 citation
Computer Science · Mathematics · #FOS: Mathematics #FOS: Physical sciences #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1909.04429

openalex publication_date 2019/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators

Cited by

• Log-Hausdorff multifractality of the absolutely continuous spectral measure of the almost Mathieu operator

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