Transport properties of kicked and quasi-periodic Hamiltonians

Details Transport properties of kicked and quasi-periodic Hamiltonians Transport properties of kicked and quasi-periodic Hamiltonians

1997-12-31

arxiv.org/abs/chao-dyn/9801003v1

Nonlinear Sciences

Chaotic Dynamics

23 pages

Scientific paper

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

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