An Agmon-Allegretto-Piepenbrink principle for Schroedinger operators

TL;DR

This paper extends the Agmon-Allegretto-Piepenbrink principle to Schrödinger operators with general potentials, providing a decomposition of the domain that characterizes the nonnegativity of the quadratic form through supersolutions.

Contribution

It introduces a novel domain decomposition approach for Schrödinger operators that links supersolutions to the nonnegativity of the associated quadratic form.

Findings

Decomposition of the domain into regions with specific properties.

Characterization of nonnegativity via supersolutions.

Extension of the principle to general Borel potentials.

Abstract

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