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

TL;DR

This paper demonstrates that certain spectral properties of the almost Mathieu operator with well-approximated frequencies can be extremely irregular, including vanishing Hausdorff measure, near-logarithmic continuity, and non-homogeneity of the spectrum.

Contribution

It provides explicit constructions of frequencies where the spectral properties of the almost Mathieu operator exhibit extreme irregularities, extending understanding of spectral behavior in this model.

Findings

Hausdorff measure of spectrum can vanish for critical coupling

Integrated density of states can have near-logarithmic modulus of continuity

Spectrum can be non-homogeneous and fail Parreau-Widom condition

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.