Date of Award
Spring 2022
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
First Advisor
Schlag, Wilhelm
Abstract
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 $\varepsilon \leq \varepsilon_0(v,\alpha)$.
Recommended Citation
Forman, Yakir Moshe, "Localization and Cantor Spectrum for Quasiperiodic Discrete Schrödinger Operators with Asymmetric, Smooth, Cosine-Like Sampling Functions" (2022). Yale Graduate School of Arts and Sciences Dissertations. 593.
https://elischolar.library.yale.edu/gsas_dissertations/593