Periodic Jacobi Matrices on Trees

TL;DR

This paper systematically studies the spectral properties of periodic Jacobi matrices on trees, establishing that these operators lack singular continuous spectrum and reviewing key prior results with new examples and open problems.

Contribution

It provides the first systematic analysis of spectral theory for periodic Jacobi matrices on trees, including a proof that they have no singular continuous spectrum.

Findings

Operators have no singular continuous spectrum

Includes review of prior results by Sunada and Aomoto

Presents new examples and open problems

Abstract

We begin the systematic study of the spectral theory of periodic Jacobi matrices on trees including a formal definition. The most significant result that appears here for the first time is that these operators have no singular continuous spectrum. We review important previous results of Sunada and Aomoto and present several illuminating examples. We present many open problems and conjectures that we hope will stimulate further work.