{
  "abstract": "We investigate a one-dimensional discrete Dirac-type operator $D_d$ acting on the Hilbert space $\\big(\\ell^2(\\mathbb{Z};\\mathbb{C})\\big)^2$, obtained from the classical continuous free Dirac operator by replacing the spatial derivative with the one-sided backward difference operator $I - S_1$, where $S_1$ denotes the right shift on $\\ell^2(\\mathbb{Z};\\mathbb{C})$. We establish the fundamental operator-theoretic and spectral properties of $D_d$: unlike symmetric or supersymmetric lattice discretizations, $D_d$ is bounded and non-normal for every non-zero rest mass $m > 0$, with self-commutator $[D_d, D_d^†] = -2mc^3(-Δ_{\\mathrm{disc}})σ_2$, yet it is quadratically normal. We compute the exact operator norm, determine the exact spectral radius in two mass regimes, and show via Fourier--Laurent analysis that the spectrum $σ(D_d)$ is purely continuous and lies on a two-component algebraic curve in the complex plane undergoing a topological transition at $mc^2 = 2c$. Finally, introducing a lattice spacing $h > 0$ and embedding $\\big(\\ell^2(h\\mathbb{Z};\\mathbb{C})\\big)^2$ isometrically into $\\big(L^2(\\mathbb{R};\\mathbb{C})\\big)^2$ via Whittaker--Shannon band-limited interpolation, we prove that the scaled non-normal discrete Dirac-type operator $D_{d,h}$ converges to the continuous self-adjoint Dirac operator $D$ in the norm resolvent sense at the optimal rate $\\mathcal{O}(h)$ as $h \\to 0^+$, without requiring frequency cutoffs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00820",
  "author_total": 1,
  "authors": [
    {
      "name": "Mariusz Niwiński"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bp7jwa9x34ded.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00820",
  "fulltext_url": null,
  "id": "bp7jwa9x34ded",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00820",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.SP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00820",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bp7jwa9x34ded.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:59Z",
      "native_id": "2611.00820",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00820"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Spectral Theory and Continuum Norm Resolvent Convergence of a One-Sided Discrete Dirac-Type Operator",
  "title_text": "Spectral Theory and Continuum Norm Resolvent Convergence of a One-Sided Discrete Dirac-Type Operator",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/bp7jwa9x34ded",
  "venue": "arXiv",
  "versions": []
}