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 Schrödinger operator −∆ + q and suitable local regularity assumptions on q, 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 q ∈ H −1 loc . 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.

Laplacians on infinite graphs: discrete vs continuous

“…The proof is straightforward and can be found in, e.g., [31,Prop. 2.21] (see also [45,Lemma 4.3]). Notice that in the case µ = ν, η coincides with the Lebesgue measure and hence ̺ η is nothing but the length metric ̺ 0 on G (see Section 2.2).…”

“…For instance, Theorem 8.3 does not imply the self-adjointness of the combinatorial Laplacian L comb when it is unbounded (see [37], [41,Theorem 6]). However, Theorems 8.1 and 8.3 enjoy a certain stability property under additive perturbations, which preserve semiboundedness ([30, Theorem 2.16], [45]). (iii) We refer for further results and details to [45], [49,Chap.…”

“…However, Theorems 8.1 and 8.3 enjoy a certain stability property under additive perturbations, which preserve semiboundedness ([30, Theorem 2.16], [45]). (iii) We refer for further results and details to [45], [49,Chap. 7.1], and [61].…”