The Density of States Measure of the Weakly Coupled Fibonacci Hamiltonian (original) (raw)

and A Gorodetski, The Density of States Measure of the Weakly Coupled Fibonacci Hamiltonian

2016

Abstract. We consider the density of states measure of the Fibonacci Hamil-tonian and show that, for small values of the coupling constant V, this measure is exact-dimensional and the almost everywhere value dV of the local scaling exponent is a smooth function of V, is strictly smaller than the Hausdorff dimension of the spectrum, and converges to one as V tends to zero. The proof relies on a new connection between the density of states measure of the Fibonacci Hamiltonian and the measure of maximal entropy for the Fibonacci trace map on the non-wandering set in the V-dependent invariant surface. This allows us to make a connection between the spectral problem at hand and the dimension theory of dynamical systems. 1.

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

Communications in Mathematical Physics, 2008

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 λ → ∞, dim(σ(H λ)) • log λ converges to an explicit constant (≈ 0.88137). We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schrödinger dynamics generated by the Fibonacci Hamiltonian.

Hyperbolicity of the trace map for the weakly coupled Fibonacci Hamiltonian

Nonlinearity, 2008

We consider the trace map associated with the Fibonacci Hamiltonian as a diffeomorphism on the invariant surface associated with a given coupling constant and prove that the non-wandering set of this map is hyperbolic if the coupling is sufficiently small. As a consequence, for these values of the coupling constant, the local and global Hausdorff dimension and the local and global box counting dimension of the spectrum of the Fibonacci Hamiltonian all coincide and are smooth functions of the coupling constant.

Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian

Communications in Mathematical Physics, 2011

We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We prove that the thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. We also show that at small coupling, all gaps allowed by the gap labeling theorem are open and the length of every gap tends to zero linearly. Moreover, for a sufficiently small coupling, the sum of the spectrum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by Even-Dar Mandel and Lifshitz. Finally, we provide explicit upper and lower bounds for the solutions to the difference equation and use them to study the spectral measures and the transport exponents. Contents

The Spectrum of the Weakly Coupled Fibonacci Hamiltonian

arXiv (Cornell University), 2009

We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We announce the following results and explain some key ideas that go into their proofs. The thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. Moreover, the length of every gap tends to zero linearly. Finally, for sufficiently small coupling, the sum of the spectrum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by Even-Dar Mandel and Lifshitz.

Spectral transitions for the square Fibonacci Hamiltonian

Journal of spectral theory, 2018

We study the spectrum and the density of states measure of the square Fibonacci Hamiltonian. We describe where the transitions from positive-measure to zero-measure spectrum and from absolutely continuous to singular density of states measure occur. This shows in particular that for almost every parameter from some open set, a positive-measure spectrum and a singular density of states measure coexist. This provides the first physically relevant example exhibiting this phenomenon.

On the Dynamical Meaning of Spectral Dimensions

Dynamical Localization theory has drawn attention to general spectral conditions which make quantum wave packet diffusion possible, and it was found that dimensional properties of the Local Density of States play a crucial role in that connection. In this paper an abstract result in this vein is presented. Time averaging over the trajectory of a wavepacket up to time T defines a statistical operator (density matrix). The corresponding entropy increases with time proportional to log T, and the coefficient of proportionality is the Hausdorff dimension of the Local Density of States, at least if the latter has good scaling properties. In more general cases, we give spectral upper and lower bounds for the increase of entropy.

Universality of fractal dimension on time-independent Hamiltonian systems

Applied Mathematics and Computation, 2009

This paper summarizes a numerical study of the dependence of the fractal dimension on the energy of certain open Hamiltonian systems, which present different kind of symmetries. Owing to the presence of chaos in these systems, it is not possible to make predictions on the way and the time of escape of the orbits starting inside the potential well. This fact causes the appearance of fractal boundaries in the initial-condition phase space. In order to compute its dimension, we use a simple method based on the perturbed orbits' behavior. The results show that the fractal dimension function depends on the structure of the potential well, contrary to other properties, such us the probability of escape, which has already been postulated as universal in earlier papers (see for instance [C. Siopis, H.E. Kandrup, G. Contopoulos, R. Dvorak, Universal properties of escape in dynamical systems, Celest. Mech. Dyn. Astr. 65 (57-68) (1997)]), from the study of Hamiltonians with different number of possible exits.