The Spectrum and the Spectral Type of the Off-Diagonal Fibonacci Operator (original) (raw)

The Spectrum of the Off-diagonal Fibonacci Operator

2010

Abstract: The family of off-diagonal Fibonacci operators can be considered as Jacobi matrices acting in. e2 (Z) with diagonal entries zero and off-diagonal entries given by sequences in the hull of the Fibonacci substitution sequence. The spectrum is independent of the sequence chosen and thus the same for all operators in the family. The spectrum is purely singular continuous and has Lebesgue measure zero. We will consider the trace map and its relation to the spectrum.

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.

Sums of regular Cantor sets of large dimension and the Square Fibonacci Hamiltonian

arXiv: Dynamical Systems, 2016

Author(s): Damanik, David; Gorodetski, Anton | Abstract: We show that under natural technical conditions, the sum of a C2C^2C2 dynamically defined Cantor set with a compact set in most cases (for almost all parameters) has positive Lebesgue measure, provided that the sum of the Hausdorff dimensions of these sets exceeds one. As an application, we show that for many parameters, the Square Fibonacci Hamiltonian has spectrum of positive Lebesgue measure, while at the same time the density of states measure is purely singular.

Dynamical maps, Cantor spectra, and localization for Fibonacci and related quasiperiodic lattices

Physical Review Letters, 1988

The one-dimensional, discrete Schrodinger equation is studied when the potential is allowed to take on two values, V~and Vg, which are arranged according to a generalized Fibonacci sequence. The problem is reduced to a dynamical map for the traces of the transfer matrices which are given recursively by Mt+& =Mt~M P, where n is a positive integer. A related class of sequences whose transfer matrices obey the recursion formula Mt~~MP-~Mt is also investigated.

Asymptotic properties of Jacobi matrices for a family of fractal measures

We study the properties and asymptotics of the Jacobi matrices associated with equilibrium measures of the weakly equilibrium Cantor sets. These family of Cantor sets were defined and different aspects of orthogonal polynomials on them were studied recently. Our main aim is numerically examine some conjectures concerning orthogonal polynomials which do not directly follow from previous results. We also compare our results with more general conjectures made for recurrence coefficients associated with fractal measures supported on mathbbR\mathbb{R}mathbbR.

Absorbing Cantor sets in dynamical systems: Fibonacci maps

1994

In this paper we shall show that there exists a polynomial unimodal map f : [0; 1] ! [0; 1] which is non-renormalizable (therefore for each x from a residual set, !(x) is equal to an interval), for which !(c) is a Cantor set and for which !(x) = !(c) for Lebesgue almost all x. So the ...

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.

Thermodynamics of fractal spectra: Cantor sets and quasiperiodic sequences

Physical Review E, 2000

We study the properties of the specific heat derived from fractal spectra, for which we extend and generalize some previous known results concerning the log-periodic oscillations of the specific heat C(T). For the monoscale case, we obtain analytically the behavior of C(T) for a two-branch general spectrum, and we show that the oscillatory regime becomes nonharmonic if there exist different gap sizes. In the multiscale case, we connect the role of the spectral dimension as the average value of C(T) with the multifractal properties of the sets, and we give a condition for which the oscillatory regime disappears. Finally, we study the thermodynamics of tight-binding Fibonacci spectra, which are not strictly invariant under changes of scale, and then many of the properties found in Cantor sets become in this case just approximated.