Sharp analysis of Maryland localization and eigenfunctions for all parameters

Maryland model is a three-parameter family of 1D quasiperiodic Schrodinger operators, that arises from a linear version of the quantum kicked rotor problem. It features hyperbolicity of the corresponding cocycle for all parameters and is known, since Simon's proof in 1985, that it has Anderson localization for all Diophantine frequencies. It was recently proved by Jitomirskaya–Liu that it has a sharp (in all parameters) spectral transition between purely singular continuous spectrum and pure point spectrum. In this talk I will present a new approach to Anderson localization in the sharp regime using the Green's function expansion, introduce the concept of phase “anti-resonance” and show how it helps us to break the conventional resonance barrier of localization. I will also describe some novel eigenfunction structures that we discover and prove using this new approach. This work is based on joint works with Svetlana Jitomirskaya and Fan Yang.