Mathematical Physics and Harmonic Analysis Seminar

Date: October 4, 2019

Time: 1:50PM - 2:50PM

Location: BLOC 628

Speaker: Wencai Liu, Texas A&M University

Title: Anderson localization for multi-frequency quasi-periodic operators on higher dimensional latices

Abstract: The first part of the talk, based on a joint work with S. Jitomirskaya and Y. Shi, is devoted to study multi-frequency quasi-periodic operators on higher dimensional lattices. We establish the Anderson localization for general analytic $k$-frequency quasi-periodic operators on $\Z^d$ for arbitrary $k, d$. This is a generalization of Bourgain-Goldstein-Schlag's result $b=d=2$ and Bourgain's result $b=d\geq 3$. Our proof works for Toeplitz operators as well. In the second part of the talk, I will discuss several closely related topics. For example, 1. Use the quantitative unique continuation to establish the Anderson localization of random Schr\"odinger operators with singular distributions. 2. Use rotation $C^{\star}$ algebra to tackle the dry ten Martini problem (gap labelling theorem). 3. Use the machinery of proof of Anderson localization to construct KAM (Kolmogorov-Arnold-Moser) tori for NLS and NLW equations.