UC Irvine

In this thesis we study discrete quasiperiodic Jacobi operators as well as ergodic operators driven by more general zero topological entropy dynamics. Such operators are deeply connected to physics (quantum Hall effect and graphene) and have enjoyed great attention from mathematics (e.g. several of Simon’s problems). The thesis has two main themes. First, to study spectral properties of quasiperiodic Jacobi matrices, in particular when off-diagonal sampling function has non-zero winding number or singularities. Second, to address the consequences of positive Lyapunov exponent for Schr\"odinger operators with a class of potentials of bounded discrepancy, prime example being those driven by shifts and skew-shifts on multi-dimensional tori.

Within the first theme, one of our results provides an if and only if topological criterion for obtaining localization from reducibility of the dual Jacobi cocycles. As an application of this result to the extended Harper’s model, we obtain sharp arithmetic spectral transition in the positive Lyapunov exponent regime. Two other results about the extended Harper’s model include a proof of non-degeneracy of all possible spectral gaps (known as Dry Ten Martini Problem) for the non-self-dual regions, and an arithmetic result on purely continuous spectrum for the self-dual region that is optimal and improves on a recent work by Avila-Jitomirskaya-Marx, who proved a measure-theoretic version.

The most important contribution among the second group is a general localization- type result for ergodic potentials of bounded discrepancy. As concrete applications of our general result, we build the first arithmetic localization-type results for potentials defined by shifts and (the first non-perturbative ones for) skew-shifts of higher- dimensional tori. Similar consideration also leads to the continuity of spectral data, for Schr\"odinger operators with such underlying dynamics in the positive Lyapunov exponent regime.