{
  "abstract": "We study a family of aperiodic metric graphs generated by one-dimensional tilings. We prove that, generically, the spectrum of the associated Kirchhoff Laplacian has Lebesgue measure zero and, apart from a discrete set of isolated eigenvalues, is a generalized Cantor set. For a large subclass of these graphs, we further show that isolated eigenvalues are generically absent. This extends several well-known results from the setting of discrete ergodic Schrödinger operators on $\\mathbb{Z}$ to a large family of metric graphs. Earlier results of this kind for metric graphs typically assume equilateral edges and exploit symmetries that reduce the analysis to a Sturm-Liouville operator on the real line. By contrast, the graph families considered here need not have equilateral edges or the symmetries needed for such a reduction and therefore require a different analytical approach.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02650",
  "author_total": 2,
  "authors": [
    {
      "name": "Ram Band"
    },
    {
      "name": "Gilad Sofer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5vr6y4e95d0dp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02650",
  "fulltext_url": null,
  "id": "5vr6y4e95d0dp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02650",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7220456208",
  "pdf_url": "https://arxiv.org/pdf/2610.02650",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5vr6y4e95d0dp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:45Z",
      "native_id": "2610.02650",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02650"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Zero measure Cantor spectrum for aperiodic metric tiling graphs",
  "title_text": "Zero measure Cantor spectrum for aperiodic metric tiling graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/5vr6y4e95d0dp",
  "venue": "arXiv",
  "versions": []
}