The Spectrum of the Weakly Coupled Fibonacci Hamiltonian

TL;DR

This paper analyzes how the spectral properties of the Fibonacci Hamiltonian evolve as the coupling constant approaches zero, revealing that the spectrum becomes increasingly thick and its Hausdorff dimension approaches one, with gaps closing linearly.

Contribution

It provides a rigorous analysis of the spectral set’s thickness and Hausdorff dimension behavior as the coupling constant tends to zero, explaining phenomena observed numerically.

Findings

Thickness tends to infinity as coupling approaches zero.

Hausdorff dimension of the spectrum tends to one.

Gaps in the spectrum close linearly with decreasing coupling.

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 announce the following results and explain some key ideas that go into their proofs. The thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. Moreover, the length of every gap tends to zero linearly. Finally, for 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.