Point Spectrum of Periodic Operators on Universal Covering Trees

TL;DR

This paper characterizes the point spectrum of operators on universal covering trees derived from finite graphs, providing a finite algorithm for computation and insights into spectral properties like delocalization.

Contribution

It offers a complete characterization of the point spectrum for pull-back operators on universal covering trees, including an efficient computation method and spectral delocalization results.

Findings

Finite algorithm to compute point spectrum from graph G

Point spectrum is contained in the spectrum of G

Typical operators exhibit spectral delocalization

Abstract

For any multi-graph $G$ with edge weights and vertex potential, and its universal covering tree $\mathcal{T}$, we completely characterize the point spectrum of operators $A_{\mathcal{T}}$ on $\mathcal{T}$ arising as pull-backs of local, self-adjoint operators $A_{G}$ on $G$. This builds on work of Aomoto, and includes an alternative proof of the necessary condition for point spectrum he derived in (Aomoto, 1991). Our result gives a finite time algorithm to compute the point spectrum of $A_{\mathcal{T}}$ from the graph $G$, and additionally allows us to show that this point spectrum is contained in the spectrum of $A_{G}$. Finally, we prove that typical pull-back operators have a spectral delocalization property: the set of edge weight and vertex potential parameters of $A_{G}$ giving rise to $A_{\mathcal{T}}$ with purely absolutely continuous spectrum is open and its complement has large codimension.