Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian

Abstract

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 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.

53 pages, 8 figures

References in corpus (5)

• The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian
• Bers and Hénon, Painlevé and Schroedinger
• Hyperbolicity of the Trace Map for the Weakly Coupled Fibonacci Hamiltonian
• Electronic Energy Spectra of Square and Cubic Fibonacci Quasicrystals
• Electronic energy spectra and wave functions on the square Fibonacci tiling

Cited by in corpus (33)

• Fractal energy spectrum of a polariton gas in a Fibonacci quasi-periodic potential
• Schrödinger Operators with Dynamically Defined Potentials: A Survey
• The Fibonacci Hamiltonian
• Revealing the topology of quasicrystals with a diffraction experiment
• Fractal dimensions of the wavefunctions and local spectral measures on the Fibonacci chain
• The fine structure of heating in a quasiperiodically driven critical quantum system
• Absolutely Continuous Convolutions of Singular Measures and an Application to the Square Fibonacci Hamiltonian
• Spectra of Discrete Schrödinger Operators with Primitive Invertible Substitution Potentials
• The fractal dimensions of the spectrum of Sturm Hamiltonian
• Nonequilibrium quench dynamics in quantum quasicrystals
• New Anomalous Lieb-Robinson Bounds in Quasi-Periodic XY Chains
• Topologically Universal Spectral Hierarchies of Quasiperiodic Systems
• Wave Functions, Quantum Diffusion, and Scaling Exponents in Golden-Mean Quasiperiodic Tilings
• Products of two Cantor sets
• Continuum Schrödinger Operators Associated With Aperiodic Subshifts
• Quantum and Spectral Properties of the Labyrinth Model
• Continuity of the spectrum of a field of self-adjoint operators
• Topological Phase Transitions from Harper to Fibonacci Crystals
• Hölder Continuity of the Integrated Density of States for the Fibonacci Hamiltonian
• Aperiodic bandgap structures for enhanced quantum two-photon sources
• Spontaneous Emission from a Fractal Vacuum
• The spectral properties of the strongly coupled Sturm Hamiltonian of eventually constant type
• On the Hausdorff dimension of the spectrum of Thue-Morse Hamiltonian
• Orthogonal polynomials of equilibrium measures supported on Cantor sets
• Hexagonal and trigonal quasiperiodic tilings
• Purely singular continuous spectrum for CMV operators generated by subshifts
• The quantum Hall effect at a weak magnetic field
• On and Off-diagonal Sturmian operator: dynamic and spectral dimension
• The spectrum of period-doubling Hamiltonian
• On the spectra of separable 2D almost Mathieu operators
• The spectral characteristics of the Sturm Hamiltonian with eventually periodic type
• Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications
• The algebraic difference of a Cantor set and its complement