Spectral analysis of tridiagonal Fibonacci Hamiltonians

TL;DR

This paper analyzes the spectral properties of Fibonacci Hamiltonians, revealing their spectrum as a zero-measure Cantor set with fractal structure, and extends existing results on these operators.

Contribution

It provides a detailed spectral analysis of tridiagonal Fibonacci Hamiltonians, including their fractal nature and Hausdorff dimension, extending previous findings.

Findings

Spectrum is a zero-measure Cantor set

Spectrum exhibits fractal structure

Results extend known properties of Fibonacci Hamiltonians

Abstract

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice, with the diagonal and the off-diagonal entries given by two sequences generated by the Fibonacci substitution on two letters. We show that the spectrum is a Cantor set of zero Lebesgue measure, and discuss its fractal structure and Hausdorff dimension. We also extend some known results on the diagonal and the off-diagonal Fibonacci Hamiltonians.