On the abominable properties of the almost Mathieu operator with well approximated frequencies

Abstract

We show that some spectral properties of the almost Mathieu operator with frequency well approximated by rationals can be as poor as at all possible in the class of all one-dimensional discrete Schroedinger operators. For the class of critical coupling, we show that the Hausdorff measure of the spectrum may vanish (for appropriately chosen frequencies) whenever the gauge function tends to zero faster than logarithmically. For arbitrary coupling, we show that modulus of continuity of the integrated density of states can be arbitrary close to logarithmic; we also prove a similar result for the Lyapunov exponent as a function of the spectral parameter. Finally, we show that (for any coupling) there exist frequencies for which the spectrum is not homogeneous in the sense of Carleson, and, moreover, fails the Parreau-Widom condition. The frequencies for which these properties hold are explicitly described in terms of the growth of the denominators of the convergents.

60 pp, 2 figures, accepted to Duke

References in corpus (5)

• The absolutely continuous spectrum of the almost Mathieu operator
• Asymptotics of spectral gaps of quasi-periodic Schrödinger operators
• Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum
• Schrödinger Operators with Thin Spectra
• Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,R)$ cocycles

Cited by in corpus (2)

• Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,R)$ cocycles
• Polynomial decay of the gap length for C^k quasi-periodic Schrodinger operators and spectral application