Quantum Dynamics via Complex Analysis Methods: General Upper Bounds Without Time-Averaging and Tight Lower Bounds for the Strongly Coupled Fibonacci Hamiltonian

We develop further the approach to upper and lower bounds in quantum dynamics via complex analysis methods which was introduced by us in a sequence of earlier papers. Here we derive upper bounds for non-time averaged outside probabilities and moments of the position operator from lower bounds for transfer matrices at complex energies. Moreover, for the time-averaged transport exponents, we present improved lower bounds in the special case of the Fibonacci Hamiltonian. These bounds lead to an optimal description of the time-averaged spreading rate of the fast part of the wavepacket in the large coupling limit. This provides the first example which demonstrates that the time-averaged spreading rates may exceed the upper box-counting dimension of the spectrum.Comment: 13 page

Scaling Estimates for Solutions and Dynamical Lower Bounds on Wavepacket Spreading

We establish quantum dynamical lower bounds for discrete one-dimensional Schr\"odinger operators in situations where, in addition to power-law upper bounds on solutions corresponding to energies in the spectrum, one also has lower bounds following a scaling law. As a consequence, we obtain improved dynamical results for the Fibonacci Hamiltonian and related models.Comment: 23 page

Power-Law Bounds on Transfer Matrices and Quantum Dynamics in One Dimension

We present an approach to quantum dynamical lower bounds for discrete one-dimensional Schr\"odinger operators which is based on power-law bounds on transfer matrices. It suffices to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamiltonian.Comment: 22 page

Power-law bounds on transfer matrices and quantum dynamics in one dimension II

We establish quantum dynamical lower bounds for a number of discrete one-dimensional Schr\"odinger operators. These dynamical bounds are derived from power-law upper bounds on the norms of transfer matrices. We develop further the approach from part I and study many examples. Particular focus is put on models with finitely or at most countably many exceptional energies for which one can prove power-law bounds on transfer matrices. The models discussed in this paper include substitution models, Sturmian models, a hierarchical model, the prime model, and a class of moderately sparse potentials.Comment: 20 page

Mixed lower bounds in quantum transport

Let H be a self-adjoint operator on a separable Hilbert space ${\cal H}$, $\psi$ some vector and ${\cal B}$ an orthonormal basis of ${\cal H}$. We consider the time-averaged moments of the position operator associated to ${\cal B}$. We derive the general lower bounds for the moments in terms of both spectral measure $\mu_\psi$ and the generalized eigenfunctions $u_\psi (n,x)$ of the state $\psi$. As a particular corollary, we generalize the recently obtained lower bound in terms of multifractal dimensions of $\mu_\psi$ and give some equivalent forms of it which can be useful in applications. In particular, we establish the relations between the $L^q$-norms ($q>1/2$) of the imaginary part of Borel transform of probability measures and the corresponding multifractal dimensions

Dynamical analysis of Schrödinger operators with growing sparse potentials

We consider Scrödinger operators in l^2(Z^+) with potentials of the form V(n)=S(n)+Q(n). Here S is a sparse potential: S(n)=n^{1-\eta \over 2 \eta}, 0<\eta <1, for n=L_N and S(n)=0 else, where L_N is a very fast growing sequence. The real function Q(n) is compactly supported. We give a rather complete description of the (time-averaged) dynamics exp(-itH) \psi for different initial states \psi. In particular, for some \psi we calculate explicitely the "intermittency function" \beta_\psi^- (p) which turns out to be nonconstant. As a particular corollary of obtained results, we show that the spectral measure restricted to (-2,2) has exact Hausdorff dimension \eta for all boundary conditions, improving the result of Jitomirskaya and Last

Transport Properties Of Markovian Anderson Model

. We consider the Anderson model in l 2 (Z d ); d 1, with potentials whose values at any site of the lattice are Markovian independent random functions of time. The upper and lower bounds for the moments jXj p (t; !) with probability= 1 are obtained. We obtain also upper and lower bounds for the averaged diffusion constant and upper bounds for the correlation function. The results present diffusive behaviour in dimensions d = 3D1; 2 up to logarithmic factors. 1 Introduction Since 50-th the Anderson model was one of the basic one for studying the transport phenomena in random environment: i @/ @t = 3D \Gamma \Delta/ +Q ! (n)/; /j t=3D0 = 3D/ 0 (n); / 0 2 l 2 (Z d ); where the values of potential Q are random i.i.d. variables. Much less attention was devoted to the models with random potentials depending on time. There are physical motivations to consider such models (see [1] - [3]), but there are almost no mathematical results (see, however, [4] for referencies). In [4] th..