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


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 (e.g., Problem 5 in B. Simon’s list of the 21st century problems) and a similar Hausdorff dimension result for the quantum graph graphene.


Svetlana Jitomirskaya

Department of Mathematics, University of California, Irvine, Irvine, CA, USA

Igor Krasovsky

Department of Mathematics, Imperial College London, London, United Kingdom