Tridiagonal substitution Hamiltonians

TL;DR

This paper studies the spectral properties of a family of Jacobi operators modulated by substitution sequences, extending methods from Schrödinger operators and analyzing their fractal and dynamical characteristics.

Contribution

It extends trace map techniques from Schrödinger to Jacobi operators with substitution modulation, revealing new spectral and fractal properties.

Findings

Analysis of spectrum and spectral type

Fractal dimensions of the spectrum

Exact dimensionality of the integrated density of states

Abstract

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the fractal structure and fractal dimensions of the spectrum, exact dimensionality of the integrated density of states, and the gap structure. We present a review of previous results, some applications, and open problems. Our investigation is based largely on the dynamics of trace maps. This work is an extension of similar results on Schroedinger operators, although some of the results that we obtain differ qualitatively and quantitatively from those for the Schoedinger operators. The nontrivialities of this extension lie in the dynamics of the associated trace map as one attempts to extend the trace map formalism from the Schroedinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are, in a sense, a test item, as many other models can be attacked via the same techniques, and we present an extensive discussion on this.