Self-adjointness of magnetic laplacians on triangulations

TL;DR

This paper extends magnetic differential operators to triangulations, introduces a geometric condition called χ-completeness that guarantees the essential self-adjointness of magnetic Laplacians, and characterizes their domains under bounded curvature conditions.

Contribution

It generalizes magnetic operators to 2-simplicial complexes and establishes conditions for their self-adjointness, advancing the understanding of magnetic Laplacians on discrete structures.

Findings

χ-completeness ensures essential self-adjointness of magnetic Laplacians

Introduction of bounded curvature hypothesis characterizes the domain of self-adjoint extensions

Extension of magnetic differential notions to triangulations compatible with gauge changes

Abstract

The notions of magnetic difference operator defined on weighted graphs or magnetic exterior derivative are discrete analogues of the notionof 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 $\chi-$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 characterize the domain of the self-adjoint extension.