SELF-ADJOINTNESS OF MAGNETIC LAPLACIANS ON TRIANGULATIONS

Abstract

The notions of magnetic difference operator or magnetic exterior derivative defined on weighted graphs are discrete analogues of the notion of covariant derivative on sections of a fibre bundle and its ex[1]tension on differential forms. In this paper, we extend these notions 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