Quantum Hamiltonians with Quasi-Ballistic Dynamics and Point Spectrum

Abstract

Consider the family of Schrödinger operators (and also its Dirac version) on $\ell^2(\mathbb{Z})$ or $\ell^2(\mathbb{N})$ \[ H^W_{ω,S}=Δ+ λF(S^nω) + W, \quad ω\inΩ, \] where $S$ is a transformation on (compact metric) $Ω$, $F$ a real Lipschitz function and $W$ a (sufficiently fast) power-decaying perturbation. Under certain conditions it is shown that $H^W_{ω,S}$ presents quasi-ballistic dynamics for $ω$ in a dense $G_δ$ 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 $H^W_{ω,S}$ with quasi-ballistic dynamics and point spectrum are also presented.