Cesar R. de Oliveira, Roberto A. Prado
Quantum Hamiltonians with Quasi-Ballistic Dynamics and Point Spectrum
(348K, pdf)
ABSTRACT. Consider the family of Schr\"o\-dinger 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.