An Agmon-Allegretto-Piepenbrink principle for Schrödinger operators.

We prove that each Borel function V : Ω → [ - ∞ , + ∞ ] defined on an open subset Ω ⊂ R N induces a decomposition Ω = S ∪ ⋃ i D i such that every function in W 0 1 , 2 ( Ω ) ∩ L 2 ( Ω ; V + d x ) 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 ∫ D i ( | ∇ ξ | 2 + V ξ 2 ) . .