{
  "abstract": "Let $(M^n,g)$ be a complete, connected, noncompact Riemannian manifold, and let $f\\ge0$ be a proper $C^2$ weight. We assume polynomial bounds on $f$, $|\\nabla f|$, and $Δf$ (Assumptions H1-H3), with growth exponent $α>0$. We study the drift Laplacian $L_f=Δ-\\langle\\nabla f,\\nabla\\cdot\\rangle$ on $L^2(M,e^{-f}dV)$. We assume that its eigenfunctions have finite polynomial growth orders $γ_k$, measured on the level sets of $f^{1/α}$. We also assume that $γ_k\\to\\infty$ in spectral order (Assumption H4). We compare this growth filtration with the filtration by eigenvalue $λ_k$. For $α>1$, we study the relation $λ_k\\asympγ_k^{(2α-2)/α}$, called compatibility. An explicit rotationally symmetric example shows that this relation can fail for the full spectrum, even when $α=2$. We prove discreteness and weighted Agmon estimates for $α>1$. A moment estimate bounds the concentration radius in terms of the growth order and a finite-scale prefactor. A lower localization condition gives one direction of the spectral comparison. For exact warped products with a one-dimensional base, we impose the radial growth condition and a regularity assumption on the radial drift. These give $α=2$ and $λ_k\\asympγ_k$ in the radial sector. For general $(M,g)$, we assume a transitive isometric symmetry of the level sets instead of a warped-product structure. The invariant-sector growth condition and radial-drift regularity then give $α=2$ and $λ_k\\asympγ_k\\asymp k$. Finally, exact polynomiality of the invariant eigenfunctions implies $α=2$ without the drift regularity assumptions. The reduced equation is then Hermite or generalized Laguerre after normalization. The arguments use neither curvature bounds nor soliton equations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25289",
  "author_total": 1,
  "authors": [
    {
      "name": "Elham Matinpour"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8n9m9mgxjr9cw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25289",
  "fulltext_url": null,
  "id": "8n9m9mgxjr9cw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25289",
  "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.DG"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7214106889",
  "pdf_url": "https://arxiv.org/pdf/2609.25289",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8n9m9mgxjr9cw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.25289",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25289"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Two Growth Filtrations for Drift Laplacians: Compatibility, Rigidity, and an Inverse Hermite/Laguerre Theorem",
  "title_text": "Two Growth Filtrations for Drift Laplacians: Compatibility, Rigidity, and an Inverse Hermite/Laguerre Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/8n9m9mgxjr9cw",
  "venue": "arXiv",
  "versions": []
}