Quantum Hamiltonians with Quasi-Ballistic Dynamics and Point Spectrum

TL;DR

This paper investigates Schrödinger and Dirac operators with dynamical potentials, demonstrating quasi-ballistic transport and point spectrum in various dynamical systems, including rotations, doubling maps, and the Anderson model.

Contribution

It establishes conditions for quasi-ballistic dynamics in these operators and provides new examples with point spectrum under rank one perturbations.

Findings

Quasi-ballistic dynamics occur for a dense G_delta set of parameters.

Applications include systems with torus rotations, doubling maps, and Axiom A systems.

Examples with point spectrum and quasi-ballistic behavior are constructed under rank one perturbations.

Abstract

Consider the family of Schr\"odinger operators (and also its Dirac version) on $\ell^2(\mathbb{Z})$ or $\ell^2(\mathbb{N})$ \[ H^W_{\omega,S}=\Delta + \lambda F(S^n\omega) + W, \quad \omega\in\Omega, \] where $S$ is a transformation on (compact metric) $\Omega$, $F$ a real Lipschitz function and $W$ a (sufficiently fast) power-decaying perturbation. Under certain conditions it is shown that $H^W_{\omega,S}$ presents quasi-ballistic dynamics for $\omega$ in a dense $G_{\delta}$ set. Applications include potentials generated by rotations of the torus with analytic condition on $F$, doubling map, Axiom A dynamical systems and the Anderson model. If $W$ is a rank one perturbation, examples of $H^W_{\omega,S}$ with quasi-ballistic dynamics and point spectrum are also presented.