Communications in Mathematical Physics

, Volume 272, Issue 1, pp 75–84 | Cite as

A Liouville-type Theorem for Schrödinger Operators

  • Yehuda PinchoverEmail author


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.


Riemannian Manifold Elliptic Operator Large Time Behavior Minimal Growth Null Sequence 
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© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsTechnion - Israel Institute of TechnologyHaifaIsrael

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