A Sears-type self-adjointness result for discrete magnetic Schr\"odinger operators

TL;DR

This paper establishes a sufficient condition for the essential self-adjointness of magnetic Schrödinger operators on weighted graphs, extending classical results from Riemannian geometry to discrete settings.

Contribution

It provides a new criterion based on metric completeness for the self-adjointness of magnetic Schrödinger operators on graphs, generalizing continuous manifold results.

Findings

Condition for self-adjointness depends on graph metric completeness.

Extends Oleinik and Shubin's results to discrete graphs.

Applicable to operators with potential functions bounded below.

Abstract

In the context of a weighted graph with vertex set $V$ and bounded vertex degree, we give a sufficient condition for the essential self-adjointness of the operator $\Delta_{\sigma}+W$, where $\Delta_{\sigma}$ is the magnetic Laplacian and $W\colon V\to\mathbb{R}$ is a function satisfying $W(x)\geq -q(x)$ for all $x\in V$, with $q\colon V\to [1,\infty)$. The condition is expressed in terms of completeness of a metric that depends on $q$ and the weights of the graph. The main result is a discrete analogue of the results of I. Oleinik and M. A. Shubin in the setting of non-compact Riemannian manifolds.