Transport Properties of Kicked and Quasiperiodic Hamiltonians

Abstract

We study transport properties of Schrödinger operators depending on one or more parameters. Examples include the kicked rotor and operators with quasi-periodic potentials. We show that the mean growth exponent of the kinetic energy in the kicked rotor and of the mean square displacement in quasiperiodic potentials is generically equal to 2: this means that the motion remains ballistic, at least in a weak sense, even away from the resonances of the models. Stronger results are obtained for a class of tight-binding Hamiltonians with an electric field E(t) = E ₀+ E ₁cos ωt. For $H = \sum {a_{n - k} (|n - k\rangle \langle n| + |n\rangle \langle n - k|) + E(t)|n\rangle \langle n|}$with $a_n\sim|n|^{-\nu}(\nu>3/2)$we show that the mean square displacement satisfies $\overline {\langle\psi_{t,}N^2\psi_1\rangle}\geqslant C_\varepsilon t^{2/(\nu+1/2)-\varepsilon}$for suitable choices of ω, E ₀, and E ₁. We relate this behavior to the spectral properties of the Floquet operator of the problem.