The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

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 $$\lambda \to \infty, {\rm dim} (\sigma(H_\lambda)) \cdot {\rm log} \lambda$$ converges to an explicit constant, $${\rm log}(1+\sqrt{2})\approx 0.88137$$ . 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.