Abstract

The development of the spectral theory of quasiperiodic operators has been largely centered around and driven by several explicit models, all coming from physics. In this talk we will review the highlights of the current state-of-the-art of the spectral theory for the following three models: almost Mathieu operator, extended Harper’s model and Maryland model, focusing on arithmetically driven spectral transitions, measure of the spectrum, and the Cantor nature of the spectrum. Those models all demonstrate interesting dependence on the arithmetics of parameters (even in some cases when the final conclusion does not have such dependence) and have traditionally been approached through KAM -type schemes. Even when the KAM arguments have been replaced by the non-perturbative ones allowing to treat more couplings, frequencies that are neither far from nor close enough to rationals presented a challenge as for them there was nothing left to perturb about. A remarkable relatively recent development concerning the explicit models is that very precise results have become possible: not only many facts have been established for a.e. frequencies and phases, but in many cases it has become possible to go deeper in the arithmetics and either establish precise arithmetic transitions or even obtain results for all values of parameters. More detailed talks on some of the covered topics will be given in the April workshop by J. You, Q. Zhou, R. Han, and W. Liu.