A Glazman–Povzner–Wienholtz theorem on graphs

The Glazman–Povzner–Wienholtz theorem states that the semiboundedness of a Schrödinger operator, when combined with suitable local regularity assumptions on its potential and the completeness of the underlying manifold, guarantees its essential self-adjointness. Our aim is to extend this result to Schrödinger operators on graphs. We first obtain the corresponding theorem for Schrödinger operators on metric graphs, allowing in particular distributional potentials which are locally H−1. Moreover, we exploit recently discovered connections between Schrödinger operators on metric graphs and weighted graphs in order to prove a discrete version of the Glazman–Povzner–Wienholtz theorem. © 2021 The Author(s)