 97654 S. De Bi\`evre, G. Forni
 Transport properties of kicked and quasiperiodic Hamiltonians
(52K, LATeX 2e)
Dec 31, 97

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Abstract. We study transport properties of Schr\"odinger operators
depending on one or more parameters. Examples include
the kicked rotor and operators with quasiperiodic 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 tightbinding Hamiltonians with an
electric field $E(t)= E_0 + E_1\cos\omega t$. For
$$
H=\sum a_{nk}(\mid nk><n\mid + \mid n>< nk\mid) + E(t)\mid n><n\mid
$$
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.
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