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.
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