Essential self-adjointness of a weighted 3-simplicial complex Laplacians

Abstract

In this paper, we construct a weighted [Formula: see text]-simplicial complex [Formula: see text] on a connected oriented locally finite graph [Formula: see text] by the introduction of the notion of oriented tetrahedrons [Formula: see text], the notion of oriented triangular faces [Formula: see text], a weight on [Formula: see text], a weight on [Formula: see text], a weight on [Formula: see text] and a weight on [Formula: see text]. Next, we create the weighted Gauss–Bonnet operator of [Formula: see text] and we use it to construct the weighted Laplacian associated to [Formula: see text], the weighted Laplacian associated to [Formula: see text], the weighted Laplacian associated to [Formula: see text], the weighted Laplacian associated to [Formula: see text] and the weighted Laplacian associated to [Formula: see text]. After that, we introduce the notion of the [Formula: see text]-completeness of [Formula: see text] and we give necessary conditions for [Formula: see text] to be [Formula: see text]-complete. Finally, we prove that the weighted Gauss–Bonnet operator and the weighted Laplacians are essentially self-adjoint based on the [Formula: see text]-completeness.