A Sears-type self-adjointness result for discrete magnetic Schrödinger operators

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 $W:V\to \mathbb{R}$ is a function satisfying $W(x)\ge -q(x)$ for all $x\in V$, with $q:V\to [1,\infty )$. 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.

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