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Communications in Mathematical Physics

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

A Liouville-type Theorem for Schrödinger Operators

  • Yehuda PinchoverEmail author
Article

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 \(\Omega\subseteq \mathbb{R}^d\) 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.

Keywords

Riemannian Manifold Elliptic Operator Large Time Behavior Minimal Growth Null Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

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

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