{
  "abstract": "We consider the one-dimensional Dirac operator $D_{A,c}$ with a general, not necessarily self-adjoint, point interaction at the origin, whose strength is described by an arbitrary complex $2\\times 2$ matrix $A$ and suitably rescaled by the speed of light $c$, together with its non-local approximations $D_{A,c}^ε$ obtained by replacing the Dirac distribution by a rank-one projection onto a scaled function $v_ε$. It is known that $D_{A,c}^ε\\to D_{A,c}$ as $ε\\to 0+$ and that $D_{A,c}-c^2/2$ converges, as $c\\to+\\infty$, to the Schrödinger operator $H_A$ with the general point interaction, both in the norm resolvent sense. We prove that the first convergence is uniform with respect to $c$ in a neighbourhood of $+\\infty$, and we provide explicit rates of convergence. Combining this with an explicit computation of the non-relativistic limit of $D_{A,c}^ε$ for fixed $ε$, we show that the two limits can be interchanged and that the corresponding double limit exists. As a by-product we obtain a family of non-local, rank-two approximations $H_A^ε$ of $H_A$ that converge to $H_A$ in the norm resolvent sense. Contrary to the relativistic setting, the coupling constants of $H_A^ε$ are renormalized; we give an explicit formula for this renormalization.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05210",
  "author_total": 1,
  "authors": [
    {
      "name": "Lukáš Heriban"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d0pd5051hv2fb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05210",
  "fulltext_url": null,
  "id": "d0pd5051hv2fb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05210",
  "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",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.SP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05210",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d0pd5051hv2fb.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:25Z",
      "native_id": "2611.05210",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05210"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Non-local approximations for the Schrödinger operator with a general point interaction via the non-relativistic limit",
  "title_text": "Non-local approximations for the Schrödinger operator with a general point interaction via the non-relativistic limit",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:38Z",
  "url": "https://research.draconitic.com/paper/d0pd5051hv2fb",
  "venue": "arXiv",
  "versions": []
}