An Agmon-Allegretto-Piepenbrink principle for Schroedinger operators

TL;DR

This work extends the Agmon-Allegretto-Piepenbrink principle to general Borel potentials by introducing a torsion-based set $S$ and decomposing $\Omega$ into Sobolev-open components $D_i$. On each component, the existence of nonnegative supersolutions yields localized, weighted Poincaré-type inequalities that control the Schrödinger form and link it to duality solutions, even for signed potentials. The authors develop a robust framework of duality solutions, an approximation scheme, and a variational setting to prove localized AAP-type results, including a construction of positive weights $w_i$ ensuring nonnegativity of the quadratic form. The results provide a principled way to understand localization phenomena and spectral properties for Schrödinger operators with highly singular or nonstandard potentials, with potential implications for PDE spectral theory and quantum mechanics. All mathematical expressions are framed within precise $\LaTeX$-style notation to support formal reasoning and downstream SEO/Indexing.

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 $\int_{D_{i}} (|\nabla ξ|^2+Vξ^2)$.

Theorems & Definitions (56)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.2
• proof : Proof of Lemma \ref{['morreyadams']}
• proof : Proof of Theorem \ref{['theoremAAPCommon']} "$\Longleftarrow$"
• Lemma 2.3
• proof : Proof of Lemma \ref{['lemmaPoincareVariational']}