A Liouville-type theorem for Schrödinger operators

2005/12/18 by Pinchover, Yehuda · 1 citation
#35B05 #35J150 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.math/0512431

Abstract

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 P_j:=-Δ+V_j, for j=0,1, are two nonnegative Schrödinger operators defined on Ω⊆ ℝ^d such that P₁ is critical in Ω with a ground state ϕ, the function ψ\nleq 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.

Cited by

• An optimal fractional Hardy inequality on the discrete half-line

Related

• A ground state alternative for singular Schrödinger operators
• A Liouville-type theorem for the p-Laplacian with potential term
• Schrodinger equations with very singular potentials in Lipschitz domains
• Shnol-type theorem for the Agmon ground state
• Uniqueness and Nondegeneracy of ground states of -Δu + (-Δ)^s u+u = u^p+1 \hboxin ℝⁿ when s is close to 0 and 1