Essential self-adjointness of the Laplacian on weighted graphs: harmonic functions, stability, characterizations and capacity

TL;DR

This paper explores conditions for the essential self-adjointness of the weighted Laplacian on graphs, introducing new characterizations involving capacity and stability, with applications to Schrödinger operators and star-like graphs.

Contribution

It provides two novel characterizations of essential self-adjointness for the Laplacian on weighted graphs, including a new capacity-based approach and stability analysis.

Findings

New capacity-based characterization of essential self-adjointness.

Extension of results to Schrödinger operators and star-like graphs.

Connections established between self-adjointness and the $ ext{l}^2$-Liouville property.

Abstract

We give two characterizations for the essential self-adjointness of the weighted Laplacian on birth-death chains. The first involves the edge weights and vertex measure and is classically known; however, we give another proof using stability results, limit point-limit circle theory and the connection between essential self-adjointness and harmonic functions. The second characterization involves a new notion of capacity. Furthermore, we also analyze the essential self-adjointness of Schr\"odinger operators, use the characterizations for birth-death chains and stability results to characterize essential self-adjointness for star-like graphs, and give some connections to the $\ell^2$-Liouville property.