Abstract
In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator P 1, such that a nonzero subsolution of a symmetric nonnegative operator P 0 is a ground state. Particularly, if P j : = −Δ + V j , for j = 0,1, are two nonnegative Schrödinger operators defined on $$\Omega\subseteq \mathbb{R}^d$$ such that P 1 is critical in Ω with a ground state φ, the function $$\psi\nleq 0$$ is a subsolution of the equation P 0 u = 0 in Ω and satisfies $$\psi_+\leq C\varphi$$ in Ω, then P 0 is critical in Ω and $$\psi$$ is its ground state. In particular, $$\psi$$ is (up to a multiplicative constant) the unique positive supersolution of the equation P 0 u = 0 in Ω. Similar results hold for general symmetric operators, and also on Riemannian manifolds.
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