Logarithmic upper bounds in quantum transport for quasi-periodic Schroedinger operators.

Math Physics Seminar
Wednesday, February 14, 2024 - 1:00pm for 1 hour (actually 50 minutes)
Skiles 006
Matthew Powell – School of Mathematics, Georgia Tech – powell@math.gatech.edu
Federico Bonetto

Please Note: Available on zoom at: https://gatech.zoom.us/j/98258240051

We shall 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 introduce the notion of the transport exponent, explain its stability, and explain how logarithmic upper bounds may be obtained in the quasi-periodic setting for all relevant parameters. This is based on joint work with S. Jitomirskaya.