On the existence and uniqueness of self-adjoint realizations of discrete (magnetic) Schr\"odinger operators

TL;DR

This paper investigates conditions under which discrete magnetic Schr"odinger operators on weighted graphs have self-adjoint realizations and when these realizations are unique, addressing fundamental questions in mathematical physics.

Contribution

It provides criteria for the existence and uniqueness of self-adjoint realizations of discrete magnetic Schr"odinger operators on weighted graphs.

Findings

Conditions for self-adjointness of the operators.

Criteria for the uniqueness of self-adjoint restrictions.

Clarification of fundamental properties of discrete magnetic Schr"odinger operators.

Abstract

In this expository paper we answer two fundamental questions concerning discrete magnetic Schr\"odinger 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.