All ETDs from UAB

Advisory Committee Chair

Gunter Stolz

Advisory Committee Members

Ryoichi Kawai

Boris Kunin

Kabe Moen

Rudi Weikard

Document Type

Dissertation

Date of Award

2021

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

We consider the discrete Schrödinger operator in $\ell^2(\Z)$ with periodic real valued potential $\mu V$ of period $p$ and $\mu>1$. Assuming that $\min\{|V(i)-V(j)|:1\le i < j\le p\}>0$ we show that the system admits a form of ballistic transport with Lieb-Robinson type bounds. In fact, we will show that the Lieb-Robinson velocity is small for large $\mu$, in fact it is proportional to $\frac{1}{\mu}$. We will also prove similar results for more general hopping operators under the same condition. In case $V$ is degenerate, i.e. there is a repeated value of $V$, we will prove weaker results for some special cases. We will also introduce another model with discrete time evolution $\tau_n(X)=(U^*)^nXU^n$ for some unitary operator $U$ and the position operator $X$. In our case the unitary operator is $U=DU_0$, where $D$ is a diagonal unitary operator. We believe that for some periodic diagonal $D$ we can make the velocity as small as we want, but we were not able to prove this. In fact, we will prove some partial results to support our thoughts.

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