On Hodge Laplacians on General Simplicial Complexes

TL;DR

This paper explores Laplacians on general simplicial complexes, establishing their self-adjointness, spectral properties, and connections to Forman curvature, thereby advancing the theoretical understanding of these operators in topological data analysis.

Contribution

It introduces formal definitions of Laplacians on countable weighted simplicial complexes and links them to Schrödinger operators and Forman curvature, providing new criteria for self-adjointness and spectral analysis.

Findings

Criteria for essential self-adjointness via Forman curvature

Gaffney type results linking completeness to spectral properties

Spectral relations between different Laplacian operators

Abstract

We study Laplacians on general countable weighted simplicial complexes from a conceptual point of view. These operators will first be introduced formally before showing that those formal operators coincide with self-adjoint realizations of operators arising from quadratic forms. A major conceptual perspective is the correspondence to signed Schr\"odinger operators unveiling the Forman curvature. The main results are criteria for essential self-adjointness via lower bounded Forman curvature and a Gaffney type result via completeness. Finally, we study spectral relations between these Laplacians.