SELF-ADJOINTNESS OF MAGNETIC LAPLACIANS ON TRIANGULATIONS COLETTE ANN ´ E, HELA AYADI, YASSIN CHEBBI, AND NABILA TORKI-HAMZA ABSTRACT. The notions of magnetic difference operator defined on weighted graphs or magnetic exterior derivative are discrete analogues of the notion of covariant derivative on sections of a fibre bundle and its extension on differential forms. In this paper, we extend this notion to certain 2-simplicial complexes called triangulations, in a manner compatible with changes of gauge. Then we study the magnetic Gauß-Bonnet operator naturally defined in this context and introduce the geometric hypothesis of χ-completeness which ensures the essential self-adjointness of this operator. This gives also the essential self-adjointness of the magnetic Laplacian on triangulations. Finally we introduce an hypothesis of bounded curvature for the magnetic potential which permits to caracterize the domain of the self-adjoint extension. 1. I NTRODUCTION Considering weighted graphs, many works are interested recently in the character of essential self- adjointness of the magnetic Laplace operator such as [11], [6], [10], [8] and [2]... Furthermore, in [1], the authors gave a new geometric criterion of χ−completeness which assures essential self-adjointness of the Laplace operator, so without discrete magnetic potential. After that, still without magnetic poten- tial, [5] generalized this notion of χ−completeness on weighted triangulations, that is a 2-simplicial complex such as the faces are all triangles. On the other hand the authors of [2], generalized this notion of χ−completeness in the case of weighted magnetic graphs, they introduced the notion of χ α −completeness related to the magnetic potential α, which is a mixing of discrete magnetic geometric properties with the behaviour of the magnetic potential. In the present work, using the analogy with the smooth case as presented in [4], we give a generalization of [1], [2] and [5] to study χ−completeness on magnetic graphs in two ways. Working on weighted triangulations, as in [5], we concider a magnetic potential α on it, ie. a 1- form with real values which has to be understood as defining a parallel transport along the edges by the quantity exp(iα). This gives a magnetic triangulation, we start with definitions for the change of gauge and the magnetic diffentials d α and its formal adjoint δ α on functions on vertices (0-forms), on skewsymmetric functions on edges (1-forms) and on skewsymmetric functions on faces (2-forms) and we study the action of a change of gauge. This magnetic differential defines naturally a Gauß-Bonnet operator T α = d α + δ α and a magnetic Laplace operator Δ α = T 2 α . We show (Theorem 1) that if the triangulation is χ−complete, in the sense of [5], then the operator T α is essentially self-adjoint, and the magnetic Laplace operator also (Corollary 7.1). We remark that this result is valid for any magnetic potential α. But the notion of magnetic potential has a geometric meaning. Using this analogy we introduce a notion of magnetic potential with bounded curvature and apply it to caracterize the domain of the self- adjoint extension of the magnetic Gauß-Bonnet operator (Theorem 2). Date: Version of May 24, 2021. 2010 Mathematics Subject Classification. 39A12, 05C63, 47B25, 05C12, 05C50. Key words and phrases. Graph, 2-Simplicial complex, Discrete magnetic operators, Essential self-adjointness, χ- completeness. 1 arXiv:2105.10171v1 [math.CO] 21 May 2021