Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian

TL;DR

This paper analyzes the spectral properties of the Fibonacci Hamiltonian as the coupling approaches zero, revealing that the spectrum's Hausdorff dimension tends to one and gaps open linearly, with implications for spectral measures and transport.

Contribution

It provides rigorous results on the spectral set’s dimension, gap structure, and sum properties at small coupling, connecting numerical phenomena to theoretical proofs.

Findings

Thickness tends to infinity as coupling approaches zero

Hausdorff dimension of the spectrum tends to one

Sum of the spectrum with itself becomes an interval at small 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 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 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.