A Glazman-Povzner-Wienholtz Theorem on graphs

TL;DR

This paper extends the Glazman-Povzner-Wienholtz theorem, originally for manifolds, to Schr"odinger operators on graphs, including metric and weighted graphs, establishing conditions for their essential self-adjointness.

Contribution

It generalizes the theorem to Schr"odinger operators on graphs with distributional potentials and connects metric and weighted graph frameworks.

Findings

Established self-adjointness criteria for Schr"odinger operators on metric graphs.

Proved a discrete version of the theorem for weighted graphs.

Extended the theorem to include distributional potentials in $H^{-1}_{ m loc}$.

Abstract

The Glazman-Povzner-Wienholtz theorem states that the completeness of a manifold, when combined with the semiboundedness of the Schr\"odinger operator $-\Delta + q$ and suitable local regularity assumptions on $q$, guarantees its essential self-adjointness. Our aim is to extend this result to Schr\"odinger operators on graphs. We first obtain the corresponding theorem for Schr\"odinger operators on metric graphs, allowing in particular distributional potentials $q\in H^{-1}_{\rm loc}$. Moreover, we exploit recently discovered connections between Schr\"odinger operators on metric graphs and weighted graphs in order to prove a discrete version of the Glazman-Povzner-Wienholtz theorem.