Geometric Criteria for Essential Self-Adjointness of Discrete Hodge Laplacians on Weighted Simplicial Complexes

TL;DR

This paper introduces geometric criteria based on $hi$-completeness to determine when discrete Hodge Laplacians on weighted simplicial complexes are essentially self-adjoint, extending understanding of their spectral properties.

Contribution

It develops the concept of $hi$-completeness for simplicial complexes and links it to essential self-adjointness of higher-dimensional Laplacians.

Findings

$hi$-completeness implies essential self-adjointness

Generalizes Laplacian analysis to higher-dimensional complexes

Provides geometric conditions for spectral analysis

Abstract

In this paper, we define the structure of $n$-simplicial complex, we consider generalizations of the Laplacians to simplicial complexes of higher dimension and we develop the notion of $\chi$-completeness for simplicial complexes. Otherwise, we study essential self-adjointness from the $\chi$-completeness geometric hypothesis.