Hodge Laplacians on Weighted Simplicial Complexes: Forms, Closures, and Essential Self-Adjointness
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914105945751552 |
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| author | Ennaceur, Marwa Jadlaoui, Amel |
| author_facet | Ennaceur, Marwa Jadlaoui, Amel |
| contents | We establish explicit operator norm bounds and essential self-adjointness criteria for discrete Hodge Laplacians on weighted graphs and simplicial complexes. For unweighted $d$-regular graphs we prove the universal estimate $\|\widetildeΔ_{1,*}\|\le 4(d-1)$, and we provide weighted extensions with a sharp comparability constant. These bounds apply without geometric completeness or curvature assumptions and ensure essential self-adjointness on natural cores. The approach extends to higher degrees via dual up/down degrees, and we show a unitary equivalence between skew and symmetric models on colorable complexes. For periodic lattices we complement the universal bounds with exact Floquet--Bloch constants, typically of order $2d$, illustrating both the sharpness in growth and the generality of our method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_15546 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hodge Laplacians on Weighted Simplicial Complexes: Forms, Closures, and Essential Self-Adjointness Ennaceur, Marwa Jadlaoui, Amel Spectral Theory Mathematical Physics Combinatorics Functional Analysis 2020: 81Q10, 47B25, 47A10, 05C63 _____________________________________________________________________ G.2.2; G.1.3; F.1.1 We establish explicit operator norm bounds and essential self-adjointness criteria for discrete Hodge Laplacians on weighted graphs and simplicial complexes. For unweighted $d$-regular graphs we prove the universal estimate $\|\widetildeΔ_{1,*}\|\le 4(d-1)$, and we provide weighted extensions with a sharp comparability constant. These bounds apply without geometric completeness or curvature assumptions and ensure essential self-adjointness on natural cores. The approach extends to higher degrees via dual up/down degrees, and we show a unitary equivalence between skew and symmetric models on colorable complexes. For periodic lattices we complement the universal bounds with exact Floquet--Bloch constants, typically of order $2d$, illustrating both the sharpness in growth and the generality of our method. |
| title | Hodge Laplacians on Weighted Simplicial Complexes: Forms, Closures, and Essential Self-Adjointness |
| topic | Spectral Theory Mathematical Physics Combinatorics Functional Analysis 2020: 81Q10, 47B25, 47A10, 05C63 _____________________________________________________________________ G.2.2; G.1.3; F.1.1 |
| url | https://arxiv.org/abs/2510.15546 |