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-k\mid) + E(t)\mid n>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.