Date of Award

Spring 2022

Document Type


Degree Name

Doctor of Philosophy (PhD)



First Advisor

Schlag, Wilhelm


This dissertation addresses the question of Anderson localization for quasi-periodic discrete Schr\"odinger operators $H = \varepsilon\Delta + V$ with potential $V$ sampled with Diophantine frequency $\alpha$ from an asymmetric, smooth, cosine-like function $v \in C^2(\mathbb{T},[-1,1])$. After reviewing the known results in case $v$ is analytic or symmetric, we develop an inductive analysis on scales, whereby we show that locally the Rellich functions of Dirichlet restrictions of $H$ inherit the cosine-like structure of $v$ and are uniformly well-separated. We utilize this construction to prove almost-sure Anderson localization and Cantor spectrum in our case for sufficiently small interaction $\varepsilon \leq \varepsilon_0(v,\alpha)$.