We exhibit an intermittency phenomenon in quantum dynamics. More precisely, we derive new lower bounds for the moments of order p associated to the state ψ(t) = e−it H ψ and averaged in time between zero and T. These lower bounds are expressed in terms of generalized fractal dimensions D ± () µψ 1/(1 + p/d) of the measure µψ (where d is the space dimension). This improves previous results obtained in terms of Hausdorff and Packing dimension. 1.

We derive a general upper bound on the spreading rate of wavepackets in the framework of Schrödinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time, where the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time independent Schrödinger equation. We also derive a new lower bound on the spreading rate, which is strongly connected with our upper bound. We apply these new bounds to the Fibonacci Hamiltonian - the most studied one-dimensional model of quasicrystals. As a result, we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization. The bounds have the same qualitative behavior in the limit of large coupling.

We develop a mathematical framework allowing to study anomalous transport in homogeneous solids. The main tools characterizing the anomalous transport properties are spectral and diffusion exponents associated to the covariant Hamiltonians describing these media. The diffusion exponents characterize the spectral measures entering in Kubo's formula for the conductivity and hence lead to anomalies in Drude's formula. We give several formulas allowing to calculate these exponents and treat, as an example, Wegner's n-orbital model as well as the Anderson model in coherent potential approximation. 1 Introduction 1.1 Anomalous electronic transport Quantum effects and interactions in various materials cause a great variety of behaviors for electronic transport at low temperature. Understanding why some materials are conductors and others insulators is a challenging central problem of solid state physics. The first attempt to get a microscopic theory of electronic transport goes back to the...

Given a positive probability Borel measure , we establish some basic properties of the associated functions (q) and of the generalized fractal dimensions D for q 2 R. We rst give the connections between the generalized fractal dimensions, the Renyi dimensions and the mean-q dimensions when q > 0. We then use these relations to prove some regularity properties for (q); we also provide some estimates for these functions (in particular estimates on their behaviour at 1), as well as for the dimensions corresponding to convolution of two measures. We nally present some calculations for speci c examples. 1

We study transport properties of Schrodinger 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 !t. For H = X an\Gammak (j n \Gamma k ?! n j + j n ?! n \Gamma k j) +E(t) j n ?! n j with an j n j \Gamma ( ? 3=2) we show that the mean square displacement satisfies ! / t ; N 2 / t ? C ffl t 2=(+1=2)\Gammaffl for suitable choices of !; E 0 and E 1 . We relate this behaviour to the spectral properties of the Floquet operator of the problem. E-mail: debievre@gat.univ-lille1.fr y E-mail: gforni@math.prince...

this paper is to contribute to the understanding of spectral localization for random Schrdinger operators in multi-dimensional Euclidean space

We study the quantum dynamics generated via Schrodinger equation by sparse-potential Jacobi matrices on l 2 (Z+ ). Exact bounds for the upper and lower intermittency functions governing the asymptotic growth of moments are derived in terms of the fractal dimensions of the spectral measure. Numerical experiments suggest that these bounds are sharp in the case of very sparse barriers. 1 Introduction We consider an evolution equation in Hilbert space H with scalar product (\Delta; \Delta): i @OE t @t = HOE t ; OE t=0 = OE 2 H: (1) H is here a self-adjoint operator with spectral resolution H = R dE . Much interest has been devoted in the last years to the following question: which information on the dynamical properties of fOE t ; t 2 Rg are encoded in the spectral measure d OE () := d(OE; E OE); (2) and--of course--in which form they can be decoded. Conventional wisdom about this problem rests on the so-called RAGE theorem [1] which provides a general qualitative answer: if the s...

We present an approach to quantum dynamical lower bounds for discrete one-dimensional Schrodinger operators which is based on power-law bounds on transfer matrices. It suces to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamiltonian.

We study a class of one-sided Hamiltonian operators with spectral measures given by invariant and ergodic measures of dynamical systems of the interval. We analyse dimensional properties of spectral measures, and prove upper bounds for the asymptotic spread in time of wavepackets. These bounds involve the Hausdorff dimension of the spectral measure, multiplied by a correction calculated from the dynamical entropy, the density of states, and the capacity of the support. For Julia matrices, the correction disappears and the growth is ruled by the fractal dimension.

Abstract not found