The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

2007/05/02 by David Damanik, Mark Embree, Anton Gorodetski +1 · 5 citations
Materials Science · Mathematics · Physics and Astronomy · #Approx #Combinatorics #Coupling constant #Dimension (graph theory) #Fibonacci number #Fractal #Geometry #Golden ratio #Hamiltonian (control theory) #Lambda #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Physics #Quantum chaos and dynamical systems #Quantum mechanics #Quasicrystal Structures and Properties #Sigma #Upper and lower bounds #math-ph #math.MP #math.SP

paper · pdf · doi:10.1007/s00220-008-0451-3

published as Commun. Math. Phys. 280 (2008), 499-516 · 23 pages

arxiv created 2007/05/02 · openalex publication_date 2008/03/03 · arxiv updated 2014/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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.

Citations

• Uniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent
• Uniform Spectral Properties of One-Dimensional Quasicrystals, III. α-Continuity

Cited by

• Hyperbolicity of the trace map for the weakly coupled Fibonacci Hamiltonian
• Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian
• On the Hausdorff Dimension of the Spectrum of the Thue–Morse Hamiltonian
• DYNAMICAL BOUNDS FOR STURMIAN SCHRÖDINGER OPERATORS
• List of fractals by Hausdorff dimension [wikipedia]

Related

• Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian
• The Fibonacci Hamiltonian
• Dynamical Upper Bounds for the Fibonacci Hamiltonian
• Uniform dynamical bounds for the Fibonacci Hamiltonian
• The fractal dimension of the spectrum of quasiperiodical schrodinger operators