{
  "abstract": "We study the asymptotic behavior of eigenvalues of nonlocal convolution-type operators $$L_\\varepsilonρ=-\\frac{1}{\\varepsilon^{d}}\\int_ΩJ\\left({\\frac{|x-y|^2}{\\varepsilon^2}}\\right)ρ(y)dy+a(x)ρ(x),$$ (with a continuous sufficiently fast decaying at infinity function $J\\geq 0$) when the scaling parameter $\\varepsilon>0$ tends to zero. Assuming that the minimum of the function $a(x)$ is attained at a point $x_0\\in \\overlineΩ\\subset \\mathbb{R}^d$ and is non-degenerate, we establish several terms in the asymptotic expansion of the first eigenvalues $λ_\\varepsilon^{(k)}$ of $L_\\varepsilon$. If $x_0$ is an interior point of the domain $Ω$, we show that $λ_\\varepsilon^{(k)}=a(x_0)-1+\\varepsilonμ^{(k)}+\\overline{o}(\\varepsilon)$, where $μ^{(k)}$ are eigenvalues of a harmonic oscillator (obtained by the blow up analysis), the corresponding eigenfunctions of $\\mathcal{L}_\\varepsilon$ are concentrated near $x_0$ on the scale $\\varepsilon^{1/2}$. If $x_0\\in\\partial Ω$ then the asymptotic expansion of the eigenvalues of $L_\\varepsilon$ has the form $λ_\\varepsilon^{(k)}=a(x_0)-1+\\varepsilon^{2/3}Λ_0+\\varepsilon(μ^{(k)}+αp_0)+\\overline{o}(\\varepsilon)$. The corresponding eigenfunctions are represented as the product of a function depending on the distance to $\\partialΩ$ and concentrated on the scale $\\varepsilon^{2/3}$, and functions localized on the scale $\\varepsilon^{1/2}$ and asymptotically independent of the distance to $\\partialΩ$. The latter functions converge (after rescaling) to eigenfunctions of a harmonic oscillator (in tangential variables), $μ^{(k)}$ being their eigenvalues. The former function is given, to the leading term, by the scaled Airy function, and $Λ_0$, $α$ and $p_0$ depend on $J(|x|^2)$ and $a(x)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13513",
  "author_total": 2,
  "authors": [
    {
      "name": "Volodymyr Rybalko"
    },
    {
      "name": "Antonina Rybalko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6hxx9a48bjdse.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13513",
  "fulltext_url": null,
  "id": "6hxx9a48bjdse",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13513",
  "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.AP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7221536567",
  "pdf_url": "https://arxiv.org/pdf/2610.13513",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6hxx9a48bjdse.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13513",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13513"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Short scale asymptotics in nonlocal spectral problems",
  "title_text": "Short scale asymptotics in nonlocal spectral problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/6hxx9a48bjdse",
  "venue": "arXiv",
  "versions": []
}