Quantum Hamiltonians with Quasi-Ballistic Dynamics and Point Spectrum

Details Quantum Hamiltonians with Quasi-Ballistic Dynamics and Point Spectrum Quantum Hamiltonians with Quasi-Ballistic Dynamics and Point Spectrum

2007-01-04

arxiv.org/abs/math-ph/0701010v1

Physics

Mathematical Physics

17 pages; to appear in Journal of Differential Equations

Scientific paper

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.

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