A Glazman-Povzner-Wienholtz Theorem on graphs
Abstract
The Glazman-Povzner-Wienholtz theorem states that the completeness of a manifold, when combined with the semiboundedness of the Schr\"odinger operator and suitable local regularity assumptions on , 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 . 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.
Keywords
Cite
@article{arxiv.2105.09931,
title = {A Glazman-Povzner-Wienholtz Theorem on graphs},
author = {Aleksey Kostenko and Mark Malamud and Noema Nicolussi},
journal= {arXiv preprint arXiv:2105.09931},
year = {2021}
}
Comments
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)