Diffusion of wave packets in a Markov random potential

We consider the evolution of a tight binding wave packet propagating in a time dependent potential. If the potential evolves according to a stationary Markov process, we show that the square amplitude of the wave packet converges, after diffusive rescaling, to a solution of a heat equation.Comment: 19 pages, acknowledgments added and typos correcte

Diffusive propagation of wave packets in a fluctuating periodic potential

We consider the evolution of a tight binding wave packet propagating in a fluctuating periodic potential. If the fluctuations stem from a stationary Markov process satisfying certain technical criteria, we show that the square amplitude of the wave packet after diffusive rescaling converges to a superposition of solutions of a heat equation.Comment: 13 pages (v2: added a paragraph on the history of the problem, added some references, correct a few typos; v3 minor corrections, added keywords and subject classes

Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees

We present examples of rooted tree graphs for which the Laplacian has singular continuous spectral measures. For some of these examples we further establish fractional Hausdorff dimensions. The singular continuous components, in these models, have an interesting multiplicity structure. The results are obtained via a decomposition of the Laplacian into a direct sum of Jacobi matrices

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as $\lambda \to \infty$, $\dim (\sigma(H_\lambda)) \cdot \log \lambda$ converges to an explicit constant ($\approx 0.88137$). We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schr\"odinger dynamics generated by the Fibonacci Hamiltonian.Comment: 23 page

Upper bounds on wavepacket spreading for random Jacobi matrices

A method is presented for proving upper bounds on the moments of the position operator when the dynamics of quantum wavepackets is governed by a random (possibly correlated) Jacobi matrix. As an application, one obtains sharp upper bounds on the diffusion exponents for random polymer models, coinciding with the lower bounds obtained in a prior work. The second application is an elementary argument (not using multiscale analysis or the Aizenman-Molchanov method) showing that under the condition of uniformly positive Lyapunov exponents, the moments of the position operator grow at most logarithmically in time.Comment: final version, to appear in CM

New characterizations of the region of complete localization for random Schr\"odinger operators

We study the region of complete localization in a class of random operators which includes random Schr\"odinger operators with Anderson-type potentials and classical wave operators in random media, as well as the Anderson tight-binding model. We establish new characterizations or criteria for this region of complete localization, given either by the decay of eigenfunction correlations or by the decay of Fermi projections. (These are necessary and sufficient conditions for the random operator to exhibit complete localization in this energy region.) Using the first type of characterization we prove that in the region of complete localization the random operator has eigenvalues with finite multiplicity

Universal Lower Bounds for Quantum Diffusion

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