{
  "abstract": "Motivated by the problem of asymptotic stability for ground states of the Klein--Gordon and Schrödinger equations in three spatial dimensions in the presence of internal modes, we study the corresponding models with cubic--quintic nonlinearities. These equations arise in several physical contexts; in particular, the cubic--quintic Klein--Gordon equation appears naturally in the study of spin-$0$ particles in quantum field theory. More precisely, let $Q_ω$ be the positive radial ground state of the three-dimensional cubic--quintic elliptic equation \\[ -ΔQ_ω+ωQ_ω-Q_ω^3+Q_ω^5=0. \\] We study two related, but logically distinct, spectral problems as $ω$ approaches the endpoint $3/16$ of the ground-state branch. First, we determine the complete discrete spectrum of the scalar Hessian \\[ L_+=-Δ+ω-3Q_ω^2+5Q_ω^4. \\] This operator is also the exact linearized spatial operator around the static state $Q_ω$ for the real scalar cubic--quintic Klein--Gordon equation with mass parameter $ω$. We prove that the number of its discrete angular-momentum sectors tends to infinity, and obtain sharp asymptotics for their locations and multiplicities. Second, for the cubic--quintic nonlinear Schrödinger equation, we analyze the full Hamiltonian matrix linearization and prove the existence of internal modes in precisely an interval of angular-momentum sectors of length comparable to $(3/16-ω)^{-1}$. %The rotating complex Klein--Gordon problem leads %to a different gyroscopically coupled pencil; since that pencil is not %analyzed here, no identification of its internal spectrum is asserted. These results provide a rigorous linear foundation for future nonlinear stability and radiation-damping analysis in three-dimensional cubic--quintic models.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16613",
  "author_total": 2,
  "authors": [
    {
      "name": "Gong Chen"
    },
    {
      "name": "Zhaojie Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/74je6vyzn7kdf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16613",
  "fulltext_url": null,
  "id": "74je6vyzn7kdf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16613",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7213399030",
  "pdf_url": "https://arxiv.org/pdf/2609.16613",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/74je6vyzn7kdf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16613",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16613"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the internal modes of the ground state for 3D cubic--quintic equations",
  "title_text": "On the internal modes of the ground state for 3D cubic--quintic equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/74je6vyzn7kdf",
  "venue": "arXiv",
  "versions": []
}