The Fibonacci Hamiltonian. (English) Zbl 1359.81108
In this paper, the authors study the Fibonacci Hamiltonian. According to the authors the spectrum of the Fibonacci Hamiltonian 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. The authors show that all the gaps of the spectrum allowed by the gap labeling theorem are open for all values of the coupling constant. Also, the authors consider 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, establish strict inequalities between them, and provide the exact large coupling asymptotics of the dimension of the density of states measure. The authors also provide the explicit relations between these spectral characteristics and the dynamical properties of the Fibonacci trace map. They establish exact identities relating the spectral and dynamical quantities, and show the connection between the spectral quantities and the thermodynamic pressure function. In the rest, the authors provide the exact statement of the results.
Reviewer: Dazmir Shulaia (Tbilisi)
MSC:
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 35P05 | General topics in linear spectral theory for PDEs |
| 82B30 | Statistical thermodynamics |
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