Mira Shamis: On the abominable properties of the Almost Mathieu operator with Liouville frequencies

Speaker
Mira Shamis, Queen Mary University of London

Abstract
This talk is devoted to the study of some spectral properties of the Almost Mathieu Operator: that is one-dimensional discrete Srchoedinger operator that acts on the space of square-summable sequences as a sum of the free discrete Laplacian and multiplication by a potential of the form \(\lambda*cos(2\pi\alpha*n + \theta)\). The parameter \(\alpha\), called the frequency, is some number between zero and one. It is well-known that the spectral properties of the Almost Mathieu operator depend sensitively on the arithmetic properties of the frequency. The case of poorly approximated frequencies that satisfy a certain Diophantine condition, is relatively well understood. In that case the spectral properties are as nice as one would expect. There is a completely different picture in the case of well approximated frequencies (Liouville numbers), in which case we show that several spectral characteristics of the Almost Mathieu operator can be as poor as at all possible in the class of all discrete Schroedinger operators. For example, the modulus of continuity of the integrated density of states (that is, of the averaged spectral measure) may be no better than logarithmic. The logarithmic modulus of continuity of the integrated density of states is known to be the optimal modulus of continuity in the class of all discrete Shroedinger operators. Other characteristics to be discussed are the Hausdorff measure of the spectrum for the so-called critical case when \(\lambda = 1\), and non-homogeneity of the spectrum (as a set) for a range of \(\lambda-s\). Based on joint work with A. Avila, Y. Last, and Q. Zhou