The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

@article{Damanik2008TheFD,
  title={The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian},
  author={David Damanik and Mark Embree and Anton Gorodetski and Serguei Tcheremchantsev},
  journal={Communications in Mathematical Physics},
  year={2008},
  volume={280},
  pages={499-516}
}
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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References

SHOWING 1-10 OF 54 REFERENCES
The spectrum of a quasiperiodic Schrödinger operator
AbstractThe spectrum σ(H) of the tight binding Fibonacci Hamiltonian (Hmn=δm,n+1+δm+1,n+δm,nμv(n),v(n)= $$\chi _{[ - \omega ^3 ,\omega ^2 [} $$ ((n−1)ω), 1/ω is the golden number) is shown to
Power law growth for the resistance in the Fibonacci model
AbstractMany one-dimensional quasiperiodic systems based on the Fibonacci rule, such as the tight-binding HamiltonianHψ(n)=ψ(n+1)+ψ(n−1)+λv(n)ψ(n),nεℤ,ψεl2(ℤ),λεℝ, wherev(n)=[(n+1)α]−[nα],[x]
Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation
AbstractA rigorous analysis is given of the dynamics of the renormalization map associated to a discrete Schrödinger operatorH onl2(ℤ), defined byHψ(n)=ψ(n+1)+ψ(n−1)+Vf(nσ)ψ(n), whereV is a real
Uniform Spectral Properties of One-Dimensional Quasicrystals, II. The Lyapunov Exponent
In this Letter we introduce a method that allows one to prove uniform local results for one-dimensional discrete Schrödinger operators with Sturmian potentials. We apply this method to the transfer
Singular continuous spectrum on a cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian
It is rigorously proven that the spectrum of the tight-binding Fibonacci Hamiltonian,Hmn=δm, n+1+δm, n−1+δm, nμ[(n+1)α]−[nα]) where α=(√5−1)/2 and [·] means integer part, is a Cantor set of zero
FRACTAL DIMENSIONS AND THE PHENOMENON OF INTERMITTENCY IN QUANTUM DYNAMICS
We exhibit an intermittency phenomenon in quantum dynamics. More precisely, we derive new lower bounds for the moments of order p associated to the state (t) = e it H and averaged in time between
Dynamical upper bounds on wavepacket spreading
We derive a general upper bound on the spreading rate of wavepackets in the framework of Schrödinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape
Hausdorff Dimension of Spectrum of One-Dimensional Schrödinger Operator with Sturmian Potentials
Let β∈(0,1) be an irrational, and [a1,a2,...] be the continued fraction expansion of β. Let Hβ be the one-dimensional Schrödinger operator with Sturmian potentials. We show that if the potential
Uniform Spectral Properties of One-Dimensional Quasicrystals, III. α-Continuity
Abstract: We study the spectral properties of one-dimensional whole-line Schrödinger operators, especially those with Sturmian potentials. Building upon the Jitomirskaya–Last extension of the
α-Continuity Properties of One-Dimensional Quasicrystals
Abstract: We apply the Jitomirskaya-Last extension of the Gilbert-Pearson theory to discrete one-dimensional Schrödinger operators with potentials arising from generalized Fibonacci sequences. We
...
1
2
3
4
5
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