On the existence and uniqueness of self-adjoint realizations of discrete (magnetic) Schrödinger operators

Abstract

In this expository paper we answer two fundamental questions concerning discrete magnetic Schrödinger operator associated with weighted graphs. We discuss when formal expressions of such operators give rise to self-adjoint operators, i.e., when they have self-adjoint restrictions. If such self-adjoint restrictions exist, we explore when they are unique.

Self-adjointness of perturbed bi-Laplacians on infinite graphs

“…It turns out that the following operator equality holds: (H| Cc(V ) ) * = H max , where the symbol T * indicates the adjoint of an operator T in ℓ 2 µ (V ). Keeping in mind that our graph is locally finite, the proof of the latter operator equality proceeds in the same way as the one for magnetic Schrödinger operators in Proposition 3.17(a) of [16].…”

“…In particular, the notion of (essential) self-adjointness has drawn quite a bit of attention; see, for instance, the papers [4,5,6,7,9,10,15,17]. The most advanced results to date concerning the self-adjointness of (primarily lower semi-bounded) magnetic Schrödinger operators (acting on vector bundles) over infinite (not necessarily locally finite) graphs are contained in the recent paper [16]. For further pointers to the literature on the self-adjointness of Laplace/Schrödinger operators on graphs, we direct the reader to [8,11,12,16].…”

“…The most advanced results to date concerning the self-adjointness of (primarily lower semi-bounded) magnetic Schrödinger operators (acting on vector bundles) over infinite (not necessarily locally finite) graphs are contained in the recent paper [16]. For further pointers to the literature on the self-adjointness of Laplace/Schrödinger operators on graphs, we direct the reader to [8,11,12,16]. Additionally, there is a line of investigation concerning Laplace-type operators acting on 1-forms, initiated by the author of [13] and continued by the authors of [1,2].…”