{
  "abstract": "We study resonance expansions and localised long-time dynamics for three-dimensional semiclassical Dirac operators. For smooth Hermitian matrix-valued perturbations which are analytic outside a compact set, we first obtain exponential cut-off resolvent bounds away from the resonance set and polynomial bounds in thin resonance-free rectangles adjacent to the real axis. Combining these estimates with a local $\\Oc(h^{-3})$ upper bound on the number of resonances, a pigeonhole selection of admissible contours, almost-analytic functional calculus, and a Cauchy--Green deformation, we derive resonance expansions for spectrally localised propagators. Positive-energy resonances contribute for large positive times, whereas negative-energy resonances contribute for large negative times. We then specialise to mass-type perturbations $\\mbfβV(x)$ in a nontrapping energy window. Using the logarithmic resonance-free region established in our earlier work together with a quantitative FBI-transform/escape-function argument, we obtain a polynomial bound for the continued cut-off resolvent and deduce rapid local-energy decay. We subsequently refine this polynomial estimate by a finite-time propagation argument in the undistorted region. Introducing the maximal positive-sheet connection time $T_χ$ associated with the spatial cutoff, we prove a geometric continued-resolvent estimate and, for every fixed $M$ below the resonance-free depth constant, derive the explicit local-energy decay threshold $T_N>T_χ+(N+1)/M$. The same analysis gives a rigorous microlocal obstruction to a uniform $[h\\log(1/h)]^{-1}$ bound for the full distorted resolvent and identifies a finite-flight-time amplification mechanism for the sandwiched continued resolvent at logarithmic depth.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02108",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhuo Chen"
    },
    {
      "name": "Michael Melgaard"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45znaz5p3mtvt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02108",
  "fulltext_url": null,
  "id": "45znaz5p3mtvt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02108",
  "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.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7216074414",
  "pdf_url": "https://arxiv.org/pdf/2610.02108",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45znaz5p3mtvt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2610.02108",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02108"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Resonance expansions and local-energy decay estimates for Dirac operators",
  "title_text": "Resonance expansions and local-energy decay estimates for Dirac operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/45znaz5p3mtvt",
  "venue": "arXiv",
  "versions": []
}