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 $P_j:=-\Delta +V_j$, for $j=0,1$, are two nonnegative Schrödinger operators defined on $\Omega \subseteq {\mathbb{R}}^d$ such that $P_1$ is critical in $\Omega$ with a ground state $\varphi$, the function $\psi \nleqq 0$ is a subsolution of the equation $P_{0}u=0$ in $\Omega$ and satisfies $|\psi |\le C\varphi$ 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 $P_{0}u=0$ in $\Omega$. Similar results hold for general symmetric operators, and also on Riemannian manifolds.

14 pages, the main result was improved, and a few more applications were added