Laplaciens de graphes infinis I Graphes m\'etriquement complets

TL;DR

This paper establishes that on metrically complete weighted graphs with bounded degree, the Laplacian and Schr"odinger operators are essentially self-adjoint, extending previous results to a broader class of graphs.

Contribution

It extends essential self-adjointness results for Laplacians and Schr"odinger operators to metrically complete weighted graphs with bounded degree.

Findings

Laplacian is essentially self-adjoint on metrically complete graphs

Schr"odinger operator is essentially self-adjoint if quadratic form is bounded below

Constructs a positive harmonic function to relate Schr"odinger operators to Laplacians

Abstract

We introduce the weighted graph Laplacian and the notion of Schr\"odinger operator on a locally finite weighted graph . Concerning essential self-adjointness, we extend Wojciechowski's and Dodziuk's results for graphs with vertex constant weight. The main result in this work states that on any metrically complete weighted graph with bounded degree, the Laplacian is essentially self-adjoint and the same holds for the Schr\"odinger operator provided the associated quadratic form is bounded from below. We construct for the proof a strictly positive and harmonic function which allows us to write any Schr\"odinger operator as a weighted graph Laplacian modulo a unitary transform.