S. De Bi\`evre, G. Forni
Transport properties of kicked and quasi-periodic Hamiltonians
(52K, LATeX 2e)

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 $\overline{<\psi_t, N^2\psi_t>}\geq
C_\epsilon t^{2/(\nu+1/2)-\epsilon}$ 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.