Transport properties of kicked and quasi-periodic Hamiltonians

Abstract

We study transport properties of Schr\"odinger 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 quasi-periodic 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 $E(t)= E_0 + E_1\cos\omega t$. For $$ H=\sum a_{n-k}(\mid n-k><n\mid + \mid n>< n-k\mid) + E(t)\mid n><n\mid $$ with $a_n\sim\mid n\mid^{-\nu} (\nu>3/2)$ we show that the mean square displacement satisfies $\bar{<\psi_t, N^2\psi_t>}\geq C_\epsilon t^{2/(\nu+1/2)-\epsilon}$ for suitable choices of $\omega, E_0$ and $E_1$. We relate this behaviour to the spectral properties of the Floquet operator of the problem.