Essential self-adjointness of non-semibounded Schrödinger operators on infinite graphs
Ognjen Milatovic
TL;DR
The paper establishes a new sufficient condition for the essential self-adjointness of discrete Schrödinger operators $\mathcal{L}_{V}$ on infinite weighted graphs that need not be locally finite or lower semi-bounded. It develops a framework using a finiteness condition (FC), an intrinsic metric $\rho$ with finite jump size and bounded degree on $\rho$-balls (condition (B*)), and a potential decomposed as $V=U-W$ with a gradient bound on $W$, to prove essential self-adjointness of $\mathcal{L}_{V}$ on $C_{c}(X)$ via an abstract perturbation theorem of Okazawa. A concrete corollary shows essential self-adjointness under a quadratic lower bound on $V$ with respect to $\rho$, i.e., $V(x)\ge -b_{1}-b_{2}[\rho(o,x)]^{2}$, when the jump size is finite and the degree-balls are bounded. The paper also presents an explicit degree-path metric example illustrating the applicability of the corollary in a non-semibounded setting, and it situates the results within broader self-adjointness methods, including Green's formula and Leibniz-type rules for discrete gradients.
Abstract
We work in the setting of infinite, not necessarily locally finite, weighted graphs. We give a sufficient condition for the essential self-adjointness of (discrete) Schrödinger operators $\mathcal{L}_{V}$ that are not necessarily lower semi-bounded. As a corollary of the main result, we show that $\mathcal{L}_{V}$ is essentially self-adjoint if the potential $V$ satisfies $V(x)\geq -b_1-b_2[ρ(0,x)]^2$, for all vertices $x$, where $o$ is a fixed vertex, $b_1$ and $b_2$ are non-negative constants, and $ρ$ is an intrinsic metric of finite jump size, such that the restriction of the weighted vertex degree to every ball corresponding to $ρ$ is bounded (not necessarily uniformly bounded).