The spectrum of the weakly coupled Fibonacci Hamiltonian

Anton Gorodetski,David Damanik

doi:10.3934/era.2009.16.23

Abstract

We consider the spectrum of the Fibonacci Hamiltonian for smallvalues of the coupling constant. It is known that this set is aCantor set of zero Lebesgue measure. Here we study the limit, as thevalue of the coupling constant approaches zero, of its thickness andits Hausdorff dimension. We announce the following results andexplain some key ideas that go into their proofs. The thicknesstends to infinity and, consequently, the Hausdorff dimension of thespectrum tends to one. Moreover, the length of every gap tends tozero linearly. Finally, for sufficiently small coupling, the sum ofthe spectrum with itself is an interval. This last result provides arigorous explanation of a phenomenon for the Fibonacci squarelattice discovered numerically by Even-Dar Mandel and Lifshitz.

Highlights

The Fibonacci Hamiltonian is a central model in the study of electronic properties of one-dimensional quasicrystals
This result was recently strengthened by Cantat [5] who showed that the Hausdorff dimension of ΣV lies strictly between zero and one
Work of Casdagli [6] and Suto [30] shows that for V ≥ 16, ΣV is a dynamically defined Cantor set. It follows from this result that the Hausdorff dimension and the upper and lower box counting dimension of ΣV all coincide; let us denote this common value by dim ΣV

Summary

Introduction

The Fibonacci Hamiltonian is a central model in the study of electronic properties of one-dimensional quasicrystals It is given by the following bounded selfadjoint operator in l2(Z),. This result was recently strengthened by Cantat [5] who showed that the Hausdorff dimension of ΣV lies strictly between zero and one. Fibonacci Hamiltonian, trace map, dynamically defined Cantor set, Hausdorff dimension, thickness. Work of Casdagli [6] and Suto [30] shows that for V ≥ 16, ΣV is a dynamically defined Cantor set It follows from this result that the Hausdorff dimension and the upper and lower box counting dimension of ΣV all coincide; let us denote this common value by dim ΣV. We would like to emphasize that quantitative properties of regular Cantor sets such as thickness and denseness are widely used in dynamical systems (see [20, 21], [24], [19]), found an application in number theory (see [16]), but to the best of our knowledge, these kinds of techniques have never been used before in the context of mathematical physics

Statement of the Results

Comments on the Proofs