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

Christopher Cedzich, Jake Fillman, Luis Velázquez

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-Velázquez (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.

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

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-Velázquez (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.
Paper Structure (7 sections, 11 theorems, 65 equations)

This paper contains 7 sections, 11 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. A Pathological Example Among ECMV Matrices
  4. Absence of Ballistic Motion
  5. A Pathological Example
  6. A Discriminant Estimate
  7. Dynamical Estimates

Key Result

Theorem 1.1

Suppose $U$ is a banded unitary operator that has only pure point spectrum and for which all eigenvectors belong to $D(X)$.In particular, this assumption is met by any banded $U$ having exponentially decaying eigenvectors. Then, for every $\psi \in D(X)$, we have

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['t.ppnoballistic']}
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 11 more