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 a 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.
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Acknowledgement
We are grateful to Mark Embree for generating the plots shown in Figs. 4 and 5.
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Communicated by G. Gallavotti
D. D. was supported in part by NSF grants DMS–0653720 and DMS–0800100.
A. G. was supported in part by NSF grant DMS–0901627.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Damanik, D., Gorodetski, A. Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian. Commun. Math. Phys. 305, 221–277 (2011). https://doi.org/10.1007/s00220-011-1220-2
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DOI: https://doi.org/10.1007/s00220-011-1220-2