A Glazman-Povzner-Wienholtz Theorem on graphs
arXiv:2105.09931 · doi:10.1016/j.aim.2021.108158
Abstract
The Glazman-Povzner-Wienholtz theorem states that the completeness of a manifold, when combined with the semiboundedness of the Schrödinger operator and suitable local regularity assumptions on , 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 . 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.
24 pages; After submission we learned that the discrete version of the Glazman-Povzner-Wienholtz theorem (Theorem 6.1) was proved earlier by a different approach in arXiv:1301.1304 (see Theorem 2.16 there)