{
  "abstract": "In this paper, we consider a broad class of continuum two-scale Hamiltonians \\begin{align*} H_\\varepsilon:=T(-i\\nabla_x+\\mathbf A(x,\\varepsilon x))+V(x,\\varepsilon x),\\qquad x\\in \\mathbb{R}^d \\end{align*} where $T$ represents either a Dirac operator or a Schrödinger operator, and $x\\mapsto \\mathbf A(x,X)$ and $x\\mapsto V(x,X)$ are $\\mathbb L$-periodic with respect to some lattice $\\mathbb L\\subset\\mathbb{R}^d$. No periodicity assumption is imposed on the second variable $X$. Let \\begin{align*} \\mathbb{R}^d\\times \\mathbb{R}^d\\ni (k,X) \\mapsto h(k,X):=T(-i\\nabla_x+k+\\mathbf A(x,X))+V(x,X) \\end{align*} be a family of operators acting on $L^2(\\mathbb{R}^d/\\mathbb{L})$ with periodic boundary conditions. We assume only local spectral information near one point $(k_0,X_0)$: an isolated $J$-dimensional Bloch bands at energy $e_0$, possibly with internal crossings, and a Hermitian matrix-valued homogeneous function of degree $m\\in\\{1,2\\}$. From these data we derive an effective Hamiltonian $\\mathfrak{h}$. Then every sufficiently localized eigenpair $(\\vec v,μ)$ of $\\mathfrak{h}$ gives an approximate eigenfunction $Θ_\\varepsilon$ such that for $\\varepsilon$ small enough, \\[ \\|Θ_\\varepsilon\\|_{L^2(\\mathbb{R}^d)}=|Ω|^{-1/2}+O(\\varepsilon^{1/2}),\\qquad \\|(H_\\varepsilon-e_0-\\varepsilon^{m/2}μ)Θ_\\varepsilon\\|_{L^2(\\mathbb{R}^d)} =O(\\varepsilon^{m/2+1/4}). \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.08320",
  "author_total": 1,
  "authors": [
    {
      "name": "Long Meng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cmebqsjscek8h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.08320",
  "fulltext_url": null,
  "id": "cmebqsjscek8h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.08320",
  "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",
      "cond-mat.mes-hall"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7167890915",
  "pdf_url": "https://arxiv.org/pdf/2607.08320",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cmebqsjscek8h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:25Z",
      "native_id": "2607.08320",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.08320"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.mes-hall",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Approximate eigenfunctions for some aperiodic crystals",
  "title_text": "Approximate eigenfunctions for some aperiodic crystals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/cmebqsjscek8h",
  "venue": "arXiv",
  "versions": []
}