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The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

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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.

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Authors and Affiliations

  1. Department of Mathematics, Rice University, Houston, TX, 77005, USA

    D. Damanik

  2. Department of Computational and Applied Mathematics, Rice University, Houston, TX, 77005, USA

    M. Embree

  3. Mathematics 253-37, California Institute of Technology, Pasadena, CA, 91125, USA

    A. Gorodetski

  4. Université d’Orléans, Laboratoire MAPMO, CNRS-UMR 6628, B.P. 6759, F-45067, Orléans Cedex, France

    S. Tcheremchantsev

Authors
  1. D. Damanik
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  2. M. Embree
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  3. A. Gorodetski
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  4. S. Tcheremchantsev
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Corresponding author

Correspondence to D. Damanik.

Additional information

Communicated by B. Simon.

D. D. was supported in part by NSF grant DMS–0653720.

M. E. was supported by NSF grant DMS-CAREER-0449973.

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Damanik, D., Embree, M., Gorodetski, A. et al. The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian. Commun. Math. Phys. 280, 499–516 (2008). https://doi.org/10.1007/s00220-008-0451-3

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  • DOI: https://doi.org/10.1007/s00220-008-0451-3

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