On the spectral theory of trees with finite cone type

TL;DR

This paper investigates the spectral properties of Laplacians on a class of rooted trees, showing that their spectra are purely absolutely continuous with finitely many bands, and analyzes the stability of this spectrum under perturbations.

Contribution

It establishes the spectral characteristics of trees with finite cone type and demonstrates stability or destruction of the absolutely continuous spectrum under specific perturbations.

Findings

Spectra are purely absolutely continuous with finitely many bands.

Small symmetric perturbations preserve the absolutely continuous spectrum in non-regular trees.

Small symmetric perturbations can destroy the absolutely continuous spectrum in regular trees.

Abstract

We study basic spectral features of graph Laplacians associated with a class of rooted trees which contains all regular trees. Trees in this class can be generated by substitution processes. Their spectra are shown to be purely absolutely continuous and to consist of finitely many bands. The main result gives stability of absolutely continuous spectrum under sufficiently small radially label symmetric perturbations for non regular trees in this class. In sharp contrast, the absolutely continuous spectrum can be completely destroyed by arbitrary small radially label symmetric perturbations for regular trees in this class.