Quantum Hamiltonians with Quasi-ballistic Dynamics and Point Spectrum

Consider the family of Schrödinger operators (and also its Dirac version) on l(Z) or l(N) H W ω,S = ∆+ λF (S n ω) +W, ω ∈ Ω, 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 ω,S presents quasi-ballistic dynamics for ω in a dense Gδ 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 ω,S with quasi-ballistic dynamics and point spectrum are also presented.