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

Abstract

We obtain a representation of the critical almost Mathieu family as a Jacobi matrix 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 short proofs of zero measure of the spectrum (e.g. Problem 5 in B. Simon's list of the 21'st century problems). We also obtain continuity of the measure of the spectrum for general singular Jacobi matrices, and prove a similar Hausdorff dimension result for the quantum graph graphene.

The authors of the original version identified a gap in Section 5 while checking the galley proofs. The affected argument has been replaced, and S. Becker has been added as a coauthor after making substantial contributions to the correction. The main results are unchanged, apart from a slight strengthening of Theorem 1.3

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