## 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 SchrÃ¶dinger 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.

## Recommended Citation

Darras, Mohammed, "Quantum velocities for one-dimensional Schrödinger operators with periodic potential." (2021). *All ETDs from UAB*. 770.

https://digitalcommons.library.uab.edu/etd-collection/770