2010/01/14 by David Damanik, Anton Gorodetski · 1 citation
Materials Science · Mathematics · #Cantor set #Coupling constant #Fibonacci number #Hamiltonian (control theory) #Hausdorff dimension #Hausdorff space #Lattice (music) #Lebesgue measure #Mathematical Dynamics and Fractals #Quantum #Quasicrystal Structures and Properties #Spectral Theory in Mathematical Physics #Spectrum (functional analysis) #math.DS #math.SP #msc:37D20 #msc:37D30 #msc:37D50 #msc:81Q10 #msc:82B44
paper · pdf · doi:10.1007/s00220-011-1220-2
published as Commun. Math. Phys. 305 (2011), 221-277 · 53 pages, 8 figures
arxiv created 2010/01/14 · openalex publication_date 2011/03/12 · arxiv updated 2015/05/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
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.