Mathematical Physics and Harmonic Analysis Seminar

Date: March 29, 2024

Time: 1:50PM - 2:50PM

Location: BLOC 302

Speaker: Matt Powell, Georgia Institute of Technology

Title: Continuity of the Lyapunov exponent for quasi-periodic Jacobi cocycles

Abstract: Many spectral properties of 1D Schr\"odinger operators have been linked to the Lyapunov exponent of the corresponding Schr\"odinger cocycle. While the situation for one-frequency quasi-periodic operators with analytic potential is well-understood, the multifrequency and non-analytic situations are not. The purpose of this talk is twofold: first, discuss our recent work on multi-frequency analytic quasi-periodic cocycles, establishing continuity (both in cocycle and jointly in cocycle and frequency) of the Lyapunov exponent for non-identically singular cocycles (of which the Jacobi cocycles form a special case), and second, discuss ongoing work extending these results to suitable Gevrey classes. Analogous results for analytic one-frequency cocycles have been known for over a decade, but the multi-frequency results have been limited to either Diophantine frequencies (continuity in cocycle) or SL(2,C) cocycles (joint continuity). We will discuss the main points of our argument, which extends earlier work of Bourgain.