Special Colloquium- Matthew Powell, Georgia Institute of Technology

Title: Quantum dynamics of ergodic Schr\”odinger operators via discrepancy. 

Abstract: Since the mid-to-late 70s, a variety of authors turned their attention to understanding the localization behavior of discrete ergodic Schr\”odinger operators. This study included the notions of Anderson localization as well as more nuanced properties of the Schr\”odinger semi-group (so-called quantum dynamics). A remarkable result of the work on the latter, due to Y. Last [1996], is that the quantum dynamics is tied to the fractal structure of the operator's spectral measures. This has been used as a suggestive indicator of certain long-time behavior of the quantum dynamics in the absence of localization. In the early 2000s, D. Damanik, S. Techeremchantsev, and others linked the long-time behavior of the quantum dynamics to properties of the Green's function of the semi-group generator, which is in turn closely related to the base dynamical system. 

In this talk, we will discuss the history of this problem, the notion of discrepancy and how it is related to ideal properties of the Green's function, and in the process, we will present current and ongoing work establishing novel upper bounds for the discrepancy for skew-shift sequences. As an application of our bounds, we establish new quantum dynamical bounds in Han-Jitomirskaya [2019], Jitomirskaya-P [2022], Shamis-Sodin [2023], and Liu [2023] for long-range Schrodinger operators with skew-shift base dynamics.