Transport properties of kicked and quasi-periodic Hamiltonians

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 $a_n\sim \mid n{\mid }^{-\nu }(\nu >3\mathrm{/}2)$ we show that the mean square displacement satisfies $\overline{<{\psi }_t,N^2{\psi }_t>}\ge C_{ϵ}{t}^{2\mathrm{/}(\nu +1\mathrm{/}2)-ϵ}$ 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.