An Agmon–Allegretto–Piepenbrink principle for Schrödinger operators

Abstract

AbstractWe prove that each Borel function$$V : \Omega \rightarrow [{-\infty }, +\infty ]$$V:Ω→[-∞,+∞]defined on an open subset$$\Omega \subset {\mathbb R}^{N}$$Ω⊂RNinduces a decomposition$$\Omega = S \cup \bigcup _{i} D_{i}$$Ω=S∪⋃iDisuch that every function in$$W^{1,2}_{0}(\Omega ) \cap L^{2}(\Omega ; V^{+} \,\mathrm {d}x)$$W01,2(Ω)∩L2(Ω;V+dx)is zero almost everywhere onSand existence of nonnegative supersolutions of$$-\Delta + V$$-Δ+Von each component$$D_{i}$$Diyields nonnegativity of the associated quadratic form$$ \int _{D_{i}} (|\nabla \xi |^2+V\xi ^2). $$∫Di(|∇ξ|2+Vξ2).