Absence of ballistic motion and presence of almost-ballistic motion for unitary operators with pure point spectrum

TL;DR

This paper demonstrates that pure point spectrum in discrete-time unitary operators prevents ballistic motion, but certain extended CMV matrices can still exhibit almost-ballistic dynamics despite having pure point spectrum.

Contribution

It adapts existing results to show pure point spectrum precludes ballistic motion and constructs examples of extended CMV matrices with pure point spectrum and near-ballistic dynamics.

Findings

Pure point spectrum prevents ballistic motion in unitary operators.

Existence of extended CMV matrices with pure point spectrum and almost-ballistic dynamics.

Sharpness of the relation between spectrum type and quantum motion.

Abstract

We adapt two results of Simon and collaborators to the setting of discrete-time unitary dynamics. We show that pure point spectrum precludes ballistic motion, and exhibit a family of examples showing that this is sharp within the class of extended Cantero--Moral-Vel\'{a}zquez (CMV) matrices: that is, there exist extended CMV matrices exhibiting pure point spectrum together with quantum dynamics as close to ballistic motion as one desires.