An Agmon-Allegretto-Piepenbrink principle for Schroedinger operators

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 $W_0^{1,2}(\Omega )\cap L^2(\Omega ;V^{+}dx)$ 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 ${\int }_{D_i}(|\nabla \xi {|}^2+V{\xi }^2)$.

References in corpus (2)

• Hopf potentials for the Schrödinger operator
• On the nonexistence of Green's function and failure of the strong maximum principle