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

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 ${\mathrm{\Delta }}_{\sigma }+W$, where ${\mathrm{\Delta }}_{\sigma }$ is the magnetic Laplacian and $W\text{\hspace{0.17em}⁣}:V\to \mathbb{R}$ is a function satisfying $W(x)\geq -q(x)$forall$x\in V$,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.