Generalized solutions and spectrum for Dirichlet forms on graphs

TL;DR

This paper explores the relationship between solutions to eigenvalue equations and the spectrum of operators associated with Dirichlet forms on infinite graphs, extending classical theorems to this setting.

Contribution

It proves generalized versions of the Allegretto-Piepenbrink and Shnol's theorems for Dirichlet forms on graphs, linking solutions and spectrum in this framework.

Findings

Positive solutions exist for energies below the spectrum's infimum.

Solutions with growth conditions imply energies are in the spectrum.

Theorems extend classical spectral results to infinite graphs.

Abstract

We study the connection of the existence of solutions with certain properties and the spectrum of operators in the framework of regular Dirichlet forms on infinite graphs. In particular we prove a version of the Allegretto-Piepenbrink theorem, which says that positive (super-)solutions to a generalized eigenvalue equation exist exactly for energies not exceeding the infimum of the spectrum. Moreover we show a version of Shnol's theorem, which says that existence of solutions satisfying a growth condition with respect to a given boundary measure implies that the corresponding energy is in the spectrum.