The Fibonacci Hamiltonian

Abstract

We consider the Fibonacci Hamiltonian, the central model in the study of electronic properties of one-dimensional quasicrystals, and provide a detailed description of its spectrum and spectral characteristics (namely, the optimal Hölder exponent of the integrated density of states, the dimension of the density of states measure, the dimension of the spectrum, and the upper transport exponent) for all values of the coupling constant (in contrast to all previous quantitative results, which could be established only in the regime of small or large coupling). In particular, we show that the spectrum of this operator is a dynamically defined Cantor set and that the density of states measure is exact-dimensional; this implies that all standard fractal dimensions coincide in each case. We show that all the gaps of the spectrum allowed by the gap labeling theorem are open for all values of the coupling constant. Also, we establish strict inequalities between the four spectral characteristics in question, and provide the exact large coupling asymptotics of the dimension of the density of states measure (for the other three quantities, the large coupling asymptotics were known before). A crucial ingredient of the paper is the relation between spectral properties of the Fibonacci Hamiltonian and dynamical properties of the Fibonacci trace map (such as dimensional characteristics of the non-wandering hyperbolic set and its measure of maximal entropy as well as other equilibrium measures, topological entropy, multipliers of periodic orbits). We establish exact identities relating the spectral and dynamical quantities, and show the connection between the spectral quantities and the thermodynamic pressure function.

47 pages

References in corpus (9)

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• Spectral Continuity for Aperiodic Quantum Systems I. General Theory
• Quantum information geometry of driven CFTs
• Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators
• Quantum and Spectral Properties of the Labyrinth Model
• The spectral properties of the strongly coupled Sturm Hamiltonian of eventually constant type
• All couplings localization for quasiperiodic operators with Lipschitz monotone potentials
• The Fibonacci quasicrystal: case study of hidden dimensions and multifractality
• Spectral properties of Schrödinger operators associated to almost minimal substitution systems
• Simple Toeplitz subshifts: combinatorial properties and uniformity of cocycles
• Transport Exponents of Sturmian Hamiltonians
• Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian
• Spectral Properties of Continuum Fibonacci Schrödinger Operators
• "Mixed spectral nature" of Thue--Morse Hamiltonian
• The spectrum of period-doubling Hamiltonian
• Spectral estimates of dynamically-defined and amenable operator families
• Aperiodic order and spectral properties
• Anderson localization for two interacting quasiperiodic particles
• The spectral characteristics of the Sturm Hamiltonian with eventually periodic type
• Finite section method for aperiodic Schrödinger operators
• Newhouse phenomena in the Fibonacci trace map
• Spectral Dimension for $\beta$-almost periodic singular Jacobi operators and the extended Harper's model
• Asymptotic property of current for a conduction model of Fermi particles on finite lattice