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

Abstract

We develop a geometric framework for the essential self-adjointness (ESA) of discrete Hodge Laplacians on weighted simplicial complexes of arbitrary dimension. Three notions of $\chi$-completeness are introduced --- \emph{global}, \emph{local by level}, and \emph{local by region} --- which are formally distinct as definitions, with the pairwise logical separations remaining partly conjectural. The main results are: \emph{(i)} ESA of the Gauss--Bonnet operator $D=d+\delta$ and the Hodge Laplacian $L=D^2$ on $\bigoplus C_c^i(\Vc)$ under operative geometric hypotheses (existence of cut-offs with finite support and bounded weighted degree), which are implied in particular by each of the three completeness notions; \emph{(ii)} ESA of the individual Laplacian block $L_{\ell }$ under the corresponding local-by-level hypotheses, via a quadratic-form argument; \emph{(iii)} ESA via a Kato--Rellich coupling decomposition under local-by-region hypotheses, applied to discrete half-spaces in the lattice ${\mathbb{Z}}^d$ ($d\ge 2$) where the coupling has operator norm $\sqrt{2}$ (graph case, i.e.\ $n=2$ in our convention; see Remark~\ref{rem:n-convention-intro}), with a sufficient condition extending the argument to $n\ge 3$; \emph{(iv)} a divergence criterion ${\sum }_k{w}_k=\infty$ guaranteeing ESA of $L$ even in the absence of $\chi$-completeness, with a worked example (a polynomial-branching tree) where ESA holds yet $\chi$-completeness fails. The relationship between the three notions is partly clarified: global $\chi$-completeness implies local $\chi$-completeness at every level (Remark~\ref{rem:hierarchy-direct}); the converse non-implications are formulated as conjectures, motivated by uniform energy lower bounds in concrete examples (Section~\ref{sec:examples}). The framework recovers and extends classical results for graphs and triangulations, including the optimal divergence rate of \cite{BGJ}.