The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs

Abstract

Abstract. The number of self-adjoint extensions of a symmetric operator acting on a complex Hilbert space is characterized by its deficiency indices. Given a locally finite unoriented simple tree, we prove that the deficiency indices of any discrete Schrödinger operator are either null or infinite. We also prove that all deterministic discrete Schrödinger operators which act on a random tree are almost surely selfadjoint. Furthermore, we provide several criteria of essential self-adjointness. We also address some importance to the case of the adjacency matrix and conjecture that, given a locally finite unoriented simple graph, its deficiency indices are either null or infinite. Besides that, we consider some generalizations of trees and weighted graphs.

Spectral and scattering theory for Schrödinger operators on perturbed topological crystals

“…The study of Laplace operators on infinite graphs has recently attracted lots of attention. Let us mention for example the problem of essential self-adjointness for very general infinite graphs [17,23], or the more precise study of the spectrum for bounded Laplacians [4,30].…”

A note on self-adjoint extensions of the Laplacian on weighted graphs

“…Recently, there has been a tremendous amount of work devoted to the study of self-adjoint extensions of certain operators defined on graphs. More specifically, these issues are studied for adjacency, (magnetic) Laplacian, and Schrödinger-type operators on graphs in [4,5,6,17,18,19,24,28,29,31,32,35,39,41,42,44,46,47,48] among others.…”

A Sears-type self-adjointness result for discrete magnetic Schrödinger operators

“…By Lemma 11 it follows that Hu ∈ ℓ 2 w (V ) and Hv ∈ ℓ 2 w (V ). Letting s → +∞ in (43) and using (23), it follows that J s → 0 as s → +∞. This, together with (42), shows (22).…”

“…An adjacency matrix operator on a locally finite graph was studied in [22]. For a study of the problem of deficiency indices for Schrödinger operators on a locally finite graph, see [23].…”