{
  "schema_version": 1,
  "problem_number": "OWR-3389-023",
  "title": "Nonmonotone Potential Means for a Convex Dirichlet Schrodinger Operator",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For the Dirichlet Schrodinger operator -d^2/dx^2 + V on (0,1), define M_j(V) as the expectation of V in the j-th L2-normalized eigenfunction. We prove M_2(t x)-M_3(t x)>t^2/20000 for every 0<t<=1. Every real bounded potential with ||V-x||_infinity<=10^-5 also has M_2(V)-M_3(V)>1/40000. In particular, V(x)=x+10^-5 x^2 is an explicit smooth strictly convex counterexample to Lavine's potential-mean monotonicity question. The proof combines the classical second-order Stark coefficient, explicitly credited to prior work, with a complex-analytic remainder bound and a quantitative stability estimate. This answers question (2) in Section 10 of OWR 2009/06, p424, also AIM Low Eigenvalues Section 3(2), p12. The other questions bundled in frozen UnsolvedMath record OWR-3389-023 remain separate; no whole-record closure is claimed. Includes a complete written proof and portable exact and symbolic checks. AI-assisted self-audited preprint; no independent peer review, formal verification or absolute-priority certification is claimed.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.SP",
    "math.CA"
  ],
  "keywords": [
    "OWR-3389-023",
    "Lavine question",
    "convex potentials",
    "Dirichlet eigenfunctions",
    "Stark effect",
    "potential expectations"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-3389-023/",
  "pdf_url": "https://eulersolve.org/papers/owr-3389-023/paper.pdf?v=a3e1e4572931",
  "doi": "10.5281/zenodo.23648929",
  "zenodo_record_url": "https://zenodo.org/records/23648929",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "On (0,1) with Dirichlet boundary conditions, V_t(x)=t x has M_2(V_t)-M_3(V_t)>t^2/20000 for every 0<t<=1. Every real bounded V with ||V-x||_infinity<=10^-5 has M_2(V)-M_3(V)>1/40000. Thus x+10^-5 x^2 is an explicit smooth strictly convex counterexample to Lavine's potential-mean monotonicity question. This refutes precisely the potential-mean monotonicity question, OWR2009/06 Section 10(2), also AIM Low Eigenvalues Section 3(2). The other questions in the bundled record remain separate. The classical Stark coefficient is credited; no absolute-priority claim is made. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "a3e1e4572931f13f9bdb4d7f88d8e4bb9e0ad56be30bf69ff4531fa0cf308269"
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    "source.zip": {
      "sha256": "2ae31f89fb3d371ce05e5cb270c90a3c2a139c87b06ca19057847133bec3c336"
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    "verification_report.md": {
      "sha256": "518d5f538f372a1a96ef0afc3d3693670df283d729d0f8be4d2a3ed6e265b508"
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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."
}