An Agmon-Allegretto-Piepenbrink principle for Schroedinger operators

Abstract

We prove that each Borel function $V : Ω\to [-\infty, +\infty]$ defined on an open subset $Ω\subset \mathbb{R}^{N}$ induces a decomposition $Ω= S \cup \bigcup_{i} D_{i}$ such that every function in $W^{1,2}_{0}(Ω) \cap L^{2}(Ω; V^{+} dx)$ is zero almost everywhere on $S$ and existence of nonnegative supersolutions of $-Δ+ V$ on each component $D_{i}$ yields nonnegativity of the associated quadratic form $\int_{D_{i}} (|\nabla ξ|^2+Vξ^2)$.