Spectral analysis for adjacency operators on graphs

TL;DR

This paper investigates the spectral properties of adjacency operators on graphs, identifying conditions under which the singular continuous spectrum is absent and providing applications to models like the 1D XY model.

Contribution

It establishes new spectral results for adjacency operators on admissible graphs, including the absence of singular continuous spectrum and eigenvalue localization at zero.

Findings

Singular continuous spectrum is absent for admissible graphs.

Eigenvalue at the origin can occur under certain conditions.

Applications include the one-dimensional XY model in physics.

Abstract

We put into evidence graphs with adjacency operator whose singular subspace is prescribed by the kernel of an auxiliary operator. In particular, for a family of graphs called admissible, the singular continuous spectrum is absent and there is at most an eigenvalue located at the origin. Among other examples, the one-dimensional XY model of solid-state physics is covered. The proofs rely on commutators methods.