A Liouville-type theorem for Schrödinger operators

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator P1, such that a nonzero subsolution of a symmetric nonnegative operator P0 is a ground state. Particularly, if Pj := −∆ + Vj , for j = 0, 1, are two nonnegative Schrödinger operators defined on Ω ⊆ R such that P1 is critical in Ω with a ground state φ, the function ψ 0 is a subsolution of the equation P0u = 0 in Ω and satisfies |ψ| ≤ Cφ in Ω, then P0 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 P0u = 0 in Ω. Similar results hold for general symmetric operators, and also on Riemannian manifolds. 2000 Mathematics Subject Classification. Primary 35J10; Secondary 35B05.