A Liouville-type theorem for Schrödinger operators

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 $Ω\subseteq \mathbb{R}^d$ such that $P_1$ is critical in $Ω$ with a ground state $ϕ$, the function $ψ\nleq 0$ is a subsolution of the equation $P_0u=0$ in $Ω$ and satisfies $|ψ|\leq Cϕ$ in $Ω$, then $P_0$ is critical in $Ω$ and $ψ$ is its ground state. In particular, $ψ$ is (up to a multiplicative constant) the unique positive supersolution of the equation $P_0u=0$ in $Ω$. Similar results hold for general symmetric operators, and also on Riemannian manifolds.