{
  "abstract": "Sharp Hardy--Rellich constants are spectral thresholds for operators with critical inverse-power potentials. Let $C_\\ell(N)$ denote the optimal constant in the $\\ell$th-order discrete Hardy--Rellich inequality on $\\Z^N$. Recent independent work established the leading high-dimensional behavior $C_\\ell(N)\\sim2^\\ell N^\\ell$ for every fixed $\\ell$. We determine the next two orders and prove \\[ C_\\ell(N)=2^\\ell N^\\ell+γ_\\ell N^{\\ell-1} +δ_\\ell N^{\\ell-2}+O_\\ell(N^{\\ell-3}), \\] with explicit coefficients $γ_\\ell$ and $δ_\\ell$. The central difficulty is a degeneracy that grows with the dimension: after conjugation, the leading operator is scalar on the $2N$ nearest neighbours of the origin. We resolve this cluster using signed-permutation symmetry and an effective operator on four lattice orbits. Weighted torus estimates and Feshbach--Schur reduction justify the finite-dimensional expansion inside the full operator, while a residual bound and Temple's inequality give the stated remainder. In particular, $C_1(N)=2N-4-20/(3N)+O(N^{-2})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30888",
  "author_total": 1,
  "authors": [
    {
      "name": "Carlos Lizama"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fywkt1pt6qvvv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30888",
  "fulltext_url": null,
  "id": "fywkt1pt6qvvv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30888",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7204868704",
  "pdf_url": "https://arxiv.org/pdf/2608.30888",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fywkt1pt6qvvv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30888",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30888"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Three-term asymptotics for discrete Hardy--Rellich constants in high dimension",
  "title_text": "Three-term asymptotics for discrete Hardy--Rellich constants in high dimension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/fywkt1pt6qvvv",
  "venue": "arXiv",
  "versions": []
}