{
  "schema_version": 1,
  "problem_number": "OWR-17290-002",
  "title": "Essential Self-Adjointness of the Laplace–Beltrami Operator for a Family of Non-Regular Almost-Riemannian Structures",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Oberwolfach Report 47/2019 records the following question, posed in the abstract of L. Rizzi's talk (joint work with V. Franceschi, D. Prandi and M. Seri): is the Laplace–Beltrami operator of the almost-Riemannian structure on R² with orthonormal frame X₁ = ∂ₓ, X₂ = x(x^{2ℓ} + z²)∂_z essentially self-adjoint on the regular region for every ℓ ≥ 1? The case ℓ = 1, and the cases ℓ ≤ n/2 of an n-dimensional version, had been settled by Prandi, Rizzi and Seri in the complete (torus) setting of their Example 7.7; their effective-potential criterion does not apply for larger ℓ. We answer the question affirmatively for every ℓ in the complete setting assumed in the report: the structures X₁ = ∂ₓ, X₂ = c(z) x(x^{2ℓ} + ρ(z))∂_z with c > 0, ρ ≥ 0 and c, cρ bounded, all of which are complete, have an essentially self-adjoint Laplace–Beltrami operator. They include structures that coincide with the model on a strip around the non-regular point; the same holds for the n-dimensional examples of Prandi, Rizzi and Seri for all ℓ and n. We do not prove a localisation theorem for arbitrary complete structures that agree with the model near that point. The proof combines a one-dimensional Hardy inequality obtained with the multiplier δ (the distance from the singular set), whose weight −δΔδ is at least 1 for this family, with a standard Agmon-type argument; the effective potential, in contrast, is not bounded below by 3/(4δ²) when ℓ ≥ 2. The same argument shows that the effective-potential hypothesis in the criteria of Prandi–Rizzi–Seri and Franceschi–Prandi–Rizzi can be replaced by −Δ_ω δ ≥ 1/δ − κ, which is an alternative sufficient condition rather than a strengthening. Taken literally on all of R², the model is incomplete, and its Laplace–Beltrami operator is not essentially self-adjoint for reasons unrelated to the singular set. The more general real-analytic conjecture of the report remains open; we give a real-analytic example without tangency points for which the weak Hardy inequality underlying all these criteria fails. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AP",
    "math.SP",
    "math.DG"
  ],
  "keywords": [
    "almost-Riemannian geometry",
    "sub-Riemannian geometry",
    "Laplace–Beltrami operator",
    "essential self-adjointness",
    "quantum confinement",
    "Hardy inequality",
    "Agmon estimate",
    "Oberwolfach Reports",
    "OWR-17290-002",
    "math.AP",
    "math.SP",
    "math.DG",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-29",
  "publication_date": "2026-09-29",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-29",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-17290-002/",
  "pdf_url": "https://eulersolve.org/papers/owr-17290-002/paper.pdf?v=2be8cb5173e4",
  "doi": "10.5281/zenodo.23041946",
  "zenodo_record_url": "https://zenodo.org/records/23041946",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers the ℓ-family question affirmatively in the complete setting assumed in the report; read literally on all of ℝ² the model is incomplete. The more general real-analytic conjecture of the report remains open. Unrefereed.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}