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

Marwa Ennaceur, Amel Jadlaoui

TL;DR

This work develops a comprehensive framework for the essential self-adjointness of discrete Hodge Laplacians on weighted simplicial complexes by introducing three notions of $\chi$-completeness (global, local-by-level, local-by-region) based on plateau cut-off functions. It proves ESA for the Gauss–Bonnet operator $D$ and the Hodge Laplacian $L$ under these geometric hypotheses, and introduces a divergence criterion that guarantees ESA even without geometric completeness in higher dimensions. The authors also establish a compact-coupling technique to handle localized defects and demonstrate that $\chi$-completeness is not necessary for ESA, providing sharp examples and counterexamples to delineate the hierarchy. Weighted adaptations show how $\chi$-completeness can be enforced or regularized via energies, making the approach applicable to topological data analysis and higher-dimensional discrete spectral theory.

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 $χ$-completeness for simplicial complexes. Otherwise, we study essential self-adjointness from the $χ$-completeness geometric hypothesis.

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

TL;DR

This work develops a comprehensive framework for the essential self-adjointness of discrete Hodge Laplacians on weighted simplicial complexes by introducing three notions of -completeness (global, local-by-level, local-by-region) based on plateau cut-off functions. It proves ESA for the Gauss–Bonnet operator and the Hodge Laplacian under these geometric hypotheses, and introduces a divergence criterion that guarantees ESA even without geometric completeness in higher dimensions. The authors also establish a compact-coupling technique to handle localized defects and demonstrate that -completeness is not necessary for ESA, providing sharp examples and counterexamples to delineate the hierarchy. Weighted adaptations show how -completeness can be enforced or regularized via energies, making the approach applicable to topological data analysis and higher-dimensional discrete spectral theory.

Abstract

In this paper, we define the structure of -simplicial complex, we consider generalizations of the Laplacians to simplicial complexes of higher dimension and we develop the notion of -completeness for simplicial complexes. Otherwise, we study essential self-adjointness from the -completeness geometric hypothesis.
Paper Structure (12 sections, 8 theorems, 70 equations)

This paper contains 12 sections, 8 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operators and closability
  4. Geometric Hypotheses and main results
  5. Geometric Hypotheses
  6. Main Results.
  7. Examples and counterexamples
  8. Examples
  9. A Detailed Example: ESA Without $\chi$-Completeness
  10. Essential Self-Adjointness of $L$
  11. Comparative study.
  12. Weighted $\chi$-Completeness and Spectral Regularization

Key Result

Lemma 3.1

For all $2 \leq i \leq n$, we have $d_i d_{i-1} = 0$ and $\delta_{i-1} \delta_i = 0$.

Theorems & Definitions (23)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Definition 4.1: Global $\chi$-Completeness
  • Definition 4.2: Local $\chi$-Completeness at Level $\ell$
  • Definition 4.3: Local $\chi$-Completeness on a Region $\Lambda$
  • Lemma 4.4
  • proof
  • Theorem 4.5
  • ...and 13 more