{
  "abstract": "We propose a rigorous computational method that certifies the separator required by Temple's inequality and its Lehmann--Goerisch refinements, and with it unconditional ground-state bounds for $H=-Δ+V$ on $R^3$ with attractive Coulomb singularities. For H$_2^+$ the method certifies $λ_2^{\\rm eee}\\ge-0.2035$ in place of the analytic $-\\tfrac14$, and Temple's inequality with a global trial then gives an enclosure of width below $6.5\\times10^{-51}$. For linear H$_3^{2+}$, whose central nucleus defeats the analytic separator, the certified widths are $1.3\\times10^{-12}$ and $3.1\\times10^{-13}$ at two nuclear separations. Such a separator, a lower bound on the second eigenvalue, is supplied by theory only for special configurations; elsewhere it is estimated and the bound is conditional. Here every hypothesis is verified in interval arithmetic at run time. Three constructions make this possible on an unbounded domain with a potential of unbounded $L^\\infty$ norm: a two-sided Neumann/Dirichlet truncation with a computable remainder; a localized Hardy inequality and sharp form bound with explicit constants, restoring coercivity; and a projection stage that certifies lower bounds on the first two eigenvalues, the second serving as separator. On a box, a Lehmann--Goerisch step places $λ_1\\in[-0.553291,-0.550740]$ for H$_2^+$. The coercivity constant, we show, must bound its own optimum from above. All drivers and certificates are released.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.25760",
  "author_total": 1,
  "authors": [
    {
      "name": "Xuefeng Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b7t5c6d4h2pce.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.25760",
  "fulltext_url": null,
  "id": "b7t5c6d4h2pce",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.25760",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7204484474",
  "pdf_url": "https://arxiv.org/pdf/2608.25760",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b7t5c6d4h2pce.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2608.25760",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.25760"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "From estimate to proof: certified ground-state energy bounds for singular Schrödinger operators on $R^3$",
  "title_text": "From estimate to proof: certified ground-state energy bounds for singular Schrödinger operators on R^3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/b7t5c6d4h2pce",
  "venue": "arXiv",
  "versions": []
}