A Liouville-type theorem for Schr\"odinger operators
  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 $P_j:=-\Delta+V_j$, for $j=0,1$, are two nonnegative Schr\"odinger operators defined on $\Omega\subseteq \mathbb{R}^d$ such that $P_1$ is critical in $\Omega$ with a ground state $\phi$, the function $\psi\nleq 0$ is a subsolution of the equation $P_0u=0$ in $\Omega$ and satisfies $|\psi|\leq C\phi$ in $\Omega$, then $P_0$ is critical in $\Omega$ and $\psi$ is its ground state. In particular, $\psi$ is (up to a multiplicative constant) the unique positive supersolution of the equation $P_0u=0$ in $\Omega$. Similar results hold for general symmetric operators, and also on Riemannian manifolds. 
Yehuda Pinchover