On Hodge Laplacians on General Simplicial Complexes
Functional Analysis
2025-08-12 v1 Metric Geometry
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.
Cite
@article{arxiv.2508.07761,
title = {On Hodge Laplacians on General Simplicial Complexes},
author = {Philipp Bartmann and Matthias Keller},
journal= {arXiv preprint arXiv:2508.07761},
year = {2025}
}