Georgia Institute of Technology College of Sciences

We will discuss the quantum dynamics associated with ergodic Schroedinger operators. Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) has been obtained for a variety of ergodic operator families, but it is well known that Anderson localization is highly unstable and can also be destroyed by generic rank one perturbations. For quasiperiodic operators, it also sensitively depends on the arithmetic properties of the phase (a seemingly irrelevant parameter from the point of view of the physics of the problem) and doesn’t hold generically. These instabilities are also present for the physically relevant notion of dynamical localization. In this talk, we will discuss the notion of discrepancy and present current and ongoing work establishing novel upper bounds of the discrepancy for skew-shift sequences. As an application of our bounds, we improve the quantum dynamical bounds in Liu [2023] and Jitomirskaya-Powell [2022].