Non-Negative Hodge–Laplace Operator

• Non-Negative Hodge–Laplace Operator is a self-adjoint, non-negative extension of the classical Laplacian, defined via coboundary and boundary operators on cochains and differential forms.
• It encodes geometric, spectral, and topological information on weighted simplicial complexes and Riemannian manifolds, with features like discrete curvature and essential self-adjointness.
• The operator underpins robust numerical schemes and functional calculus frameworks, facilitating advances in spectral geometry and finite element exterior calculus.

The non-negative Hodge–Laplace operator generalizes the classical Laplacian from differential topology to abstract settings such as weighted simplicial complexes, discrete graphs, and Riemannian manifolds. Defined via the coboundary and boundary operators acting between spaces of $k$-cochains or differential forms, and typically realized as a self-adjoint, non-negative extension from a canonical core, the non-negative Hodge–Laplace operator is a central object in topological, analytic, and numerical analysis. On combinatorial and manifold settings alike, the structure of these operators encodes geometric, spectral, and topological information, including discrete curvature (e.g., Forman curvature) and ensures key analytic properties such as essential self-adjointness and functional calculus frameworks.

1. Formal Definition on Weighted Simplicial Complexes

Let $\Sigma$ be a countable simplicial complex equipped with strictly positive weights $m\colon\Sigma\to(0,\infty)$. For each $k\ge0$, denote $\Sigma_k$ as the set of $k$-simplices and consider the real Hilbert space of $k$-cochains

$$\ell^2(\Sigma_k,m)=\bigg\{\,f:\Sigma_k\to\mathbb{R} \;\bigg|\; \sum_{\sigma\in\Sigma_k} m(\sigma)|f(\sigma)|^2<\infty\,\bigg\},$$

with inner product $$\langle f,g\rangle = \sum_{\sigma\in\Sigma_k} m(\sigma)f(\sigma)g(\sigma)$$. The coboundary operator

$$d_k:\ell^2(\Sigma_k)\to\ell^2(\Sigma_{k+1}), \quad \big(d_k f\big)(\sigma) = \sum_{\tau\prec\sigma} \theta(\tau,\sigma)f(\tau),$$

with orientation $\theta(\tau,\sigma)=\pm1$, satisfies $d_k\circ d_{k-1}=0$. Its formal adjoint (the boundary operator) is

$$\partial_k:\ell^2(\Sigma_k)\to\ell^2(\Sigma_{k-1}), \quad \partial_kf(\tau) = \frac{1}{m(\tau)}\sum_{\sigma\succ\tau} m(\sigma)\theta(\tau,\sigma)f(\sigma).$$

The $k$-th non-negative Hodge Laplacian is defined on compactly supported cochains as

$$\Delta_k := d_k^*d_k + d_{k-1}d_{k-1}^* = \partial_{k+1}d_k + d_{k-1}\partial_k.$$

One verifies that $\Delta_k$ is symmetric and non-negative on this core, and under suitable conditions (detailed below), extends uniquely to a self-adjoint, non-negative operator on $\ell^2(\Sigma_k,m)$ (Bartmann et al., 11 Aug 2025, Ennaceur et al., 17 Oct 2025).

2. Quadratic Forms, Self-Adjointness, and Domain Characterization

The Hodge–Laplace operator $\Delta_k$ is associated to the closed, densely defined quadratic form

$Q_k(f) = \|d_kf\|^2 + \|\partial_k f\|^2,$

with form domain

$$D(Q_k) = \{f\in\ell^2(\Sigma_k): d_kf\in\ell^2,\ \partial_k f\in\ell^2\}.$$

Stokes’ theorem (discrete Green’s formula) ensures $Q_k$ is closed and non-negative. By the representation theorem for quadratic forms, this yields a unique self-adjoint realization $\Delta_k\ge0$ such that

$$Q_k(f,g) = \langle f,\Delta_kg\rangle, \quad \forall f\in D(Q_k),\, g\in D(\Delta_k)\subset D(Q_k).$$

On finite or sufficiently rapidly decaying cochains, $$\Delta_kf = \partial_{k+1}d_kf + d_{k-1}\partial_kf$$ holds pointwise. Essential self-adjointness of the minimal operator (defined on compactly supported cochains) is ensured under various geometric and curvature conditions (see Section 4) (Bartmann et al., 11 Aug 2025, Ennaceur et al., 17 Oct 2025).

3. Schrödinger Representation, Forman Curvature, and Discrete Spectral Geometry

The non-negative Hodge–Laplace operator can be rewritten as a (signed) discrete Schrödinger operator,

$$(\Delta_k f)(\tau) = \frac{1}{m(\tau)}\sum_{\tau'\sim\tau} b(\tau,\tau')\big(f(\tau)-o(\tau,\tau')f(\tau')\big) + \frac{c^k(\tau)}{m(\tau)}f(\tau),$$

where $b(\tau,\tau')>0$ encodes "hopping", $o(\tau,\tau')=\pm1$ records orientations, and $c^k(\tau)$ is the Forman curvature at $\tau$. The explicit combinatorial Forman curvature formula is

$$c^k(\tau) = \sum_{\rho\prec\tau}\frac{m(\tau)^2}{m(\rho)} + \sum_{\sigma\succ\tau}m(\sigma) - \sum_{\substack{\tau'\neq\tau,\, \tau\wedge\tau'\neq\emptyset}}\Bigl|\frac{m(\tau)m(\tau')}{m(\tau\wedge\tau')} - \sum_{\sigma\succ\tau,\sigma\succ\tau'}m(\sigma)\Bigr|$$

For uniform weights ($m\equiv1$), this reduces to a combinatorial sum of the numbers of faces, cofaces, and shared coface links.

A lower bound $c^k(\tau)\ge -K m(\tau)$ for all $\tau\in\Sigma_k$ ensures that $\Delta_k$ is bounded below by $-K$, which implies non-negativity of the Friedrichs extension and essential self-adjointness by Agmon–Shnol–Faris–Milch criteria (Bartmann et al., 11 Aug 2025).

4. Metric Completeness, Essential Self-Adjointness, and Up/Down Interlacing

Constructing an intrinsic metric $d$ on the 1-skeleton of $\Sigma_k$ such that

$$\sum_{\tau'} b(\tau,\tau') d(\tau,\tau')^2 \le m(\tau)$$

enables a Gaffney-type completeness result: if $(\Sigma_k,d)$ is metrically complete (all closed $d$-balls are finite or compact), then the minimal Hodge Laplacian is essentially self-adjoint. This extends an analogue of Gaffney’s theorem for complete Riemannian manifolds to the discrete combinatorial setting. The up/down structure of the Hodge Laplacian leads to spectral interlacing: $$\sigma(\Delta_k^{up})\setminus\{0\} = \sigma(\Delta_{k+1}^{down})\setminus\{0\}$$ and for the full Hodge Laplacian $\Delta_k^H$,

$$\sigma(\Delta_k^H)\setminus\{0\} = \sigma(\Delta_k^{up})\setminus\{0\} = \sigma(\Delta_k^{down})\setminus\{0\}$$

under the coincidence of form domains (Bartmann et al., 11 Aug 2025).

5. Explicit Norm Bounds, Boundedness, and Examples

On weighted finite simplicial complexes, define down-degree and up-degree as

$$D_k^\downarrow = \sup_{\tau\in T_{k-1}}\frac{1}{m_{k-1}(\tau)}\sum_{\sigma\supset\tau} m_k(\sigma), \qquad D_k^\uparrow = \sup_{\sigma\in T_k}\frac{1}{m_k(\sigma)}\sum_{\upsilon\supset\sigma} m_{k+1}(\upsilon).$$

For the normalized operator $\widetilde\Delta_k$ on unweighted $\ell^2$,

$$\|\widetilde\Delta_k\|\le D_k^\downarrow+D_k^\uparrow$$

with $\Delta_k$ bounded whenever the total degree is finite. For $d$-regular unweighted graphs,

$$D_1^\downarrow = D_1^\uparrow = 2(d-1),\quad \|\widetilde\Delta_1\|\le 4(d-1).$$

In the square lattice ($d=4$), $\|\widetilde\Delta_1\|\le12$; the exact value for this case is $8$ (via Floquet–Bloch analysis). For top degree $k=n$, the operator reduces to a weighted adjacency plus diagonal potential, with norm bounded by the sum of adjacency kernel and potential maximum (Ennaceur et al., 17 Oct 2025).

Table: Operator Norm Bounds in Lattice Examples

Lattice	Degree $d$	Universal Bound $4(d-1)$	$\|\widetilde\Delta_1\|$ (Exact)
$\mathbb{Z}^2$	4	12	8
Triangular	6	20	12
$\mathbb{Z}^3$	6	20	12
BCC	8	28	16
FCC	12	44	$\approx$24

These explicit operator bounds ensure essential self-adjointness and spectral boundedness in periodic and weighted settings (Ennaceur et al., 17 Oct 2025).

6. Analytic and Functional Calculus Properties on Manifolds

For the weighted Hodge Laplacian (Witten Laplacian) on a complete Riemannian manifold $(M,g)$ with smooth weight $p$, the operator

$L_{k,m} = d_{k,m}^*d_{k,m} + d_{k-1,m}d_{k-1,m}^*$

is considered on weighted $L^2$-spaces of $k$-forms. Under the hypothesis of non-negative Bakry–Émery Ricci curvature $Q_k\ge0$, $L_{k,m}$ is non-negative and self-adjoint; it generates a symmetric contraction semigroup and satisfies a Bochner–Weitzenböck formula controlling curvature terms. The Hodge–Dirac operator $D_m = d_m + d_m^*$ satisfies $D_m^2 = L_{0,m} \oplus \cdots \oplus L_{n,m}$.

For all $1<p<\infty$, $L_{k,m}$ is R-sectorial with a bounded $H^\infty$-functional calculus on $L^p(A^k TM, m)$; the associated heat semigroup exhibits Gaussian off-diagonal decay. R-bisectoriality and functional calculus also extend to the Hodge–Dirac operator on full exterior bundles (Neerven et al., 2017).

7. Numerical Schemes and Discretizations

Structure-preserving discretizations, such as those arising from finite element exterior calculus (FEEC) and its broken/nonconforming variants (e.g., CONGA method), yield discrete non-negative Hodge–Laplace operators with analogous kernel (harmonic field) and spectral properties, provided essential assumptions (bounded cochain projections, moment preservation, symmetric stabilization) are satisfied. The discrete operator on the broken FEEC space,

$$L^\ell_{h,\alpha} := d^{\ell-1}_h \delta^\ell_h + \delta^{\ell+1}_h d^\ell_h + \alpha(I-P^\ell_h)^*(I-P^\ell_h)$$

is self-adjoint, non-negative, spectrally correct, and preserves convergence and eigenstructure under standard regularity and stability hypotheses. The kernel coincides with that of the conforming FEEC discretized complex, ensuring no spurious harmonic fields (Campos-Pinto et al., 2021).

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In summary, the non-negative Hodge–Laplace operator, rigorously realized as a self-adjoint, non-negative extension from a canonical core on spaces of cochains or forms, encodes deep connections among topology, geometry, and analysis, both in discrete and smooth settings. Essential self-adjointness is governed by geometric completeness and curvature bounds (e.g., Forman curvature in the discrete, Bakry–Émery Ricci in the smooth), with robust analytic and spectral features inherited via operator-theoretic frameworks and their discrete analogues (Bartmann et al., 11 Aug 2025, Ennaceur et al., 17 Oct 2025, Campos-Pinto et al., 2021, Neerven et al., 2017).