The Spectrum and the Spectral Type of the Off-Diagonal Fibonacci Operator

TL;DR

This paper investigates the spectral properties of Fibonacci-based Jacobi matrices, revealing that their spectrum has zero Lebesgue measure, is purely singular continuous, and can form a dynamically defined Cantor set under certain conditions.

Contribution

It provides new insights into the spectral nature of Fibonacci off-diagonal operators, including measure, continuity, and fractal structure, extending understanding of quasicrystal models.

Findings

Spectrum has zero Lebesgue measure

Spectral measures are purely singular continuous

Spectrum forms a Cantor set under certain parameter conditions

Abstract

We consider Jacobi matrices with zero diagonal and off-diagonals given by elements of the hull of the Fibonacci sequence and show that the spectrum has zero Lebesgue measure and all spectral measures are purely singular continuous. In addition, if the two hopping parameters are distinct but sufficiently close to each other, we show that the spectrum is a dynamically defined Cantor set, which has a variety of consequences for its local and global fractal dimension.