{
  "schema_version": 1,
  "problem_number": "OWR-15951-003",
  "title": "On Fillman's Question for Separable Continuum Fibonacci Operators: Half-Lines for Two Couplings and the High-Energy Fricke–Vogt Invariant",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "J. Fillman asked (Oberwolfach Report 46/2017) whether, for the continuum Fibonacci potential V_ω built from two bump functions f_0, f_1 ∈ L²[0,1) on unit cells, the spectrum of the separable operator −Δ + λ_1 V_ω(x) + λ_2 V_ω(y) on L²(ℝ²), which is the sum σ(L_{λ_1 V_ω}) + σ(L_{λ_2 V_ω}) of two one-dimensional spectra, contains a half-line for all λ_1, λ_2 > 0. We give a partial answer. For locally constant bumps f_j = c_j χ_[0,1) the answer is yes for all real λ_1, λ_2 (Theorem A). For λ_1 = λ_2 this is a theorem of Damanik, Fillman and Gorodetski (J. Funct. Anal. 2021); our proof uses their Proposition 4.2 and their construction of curve pieces as their proof does, together with a two-set version of their abstract Bethe–Sommerfeld lemma; the windows that carry thick Cantor parts of the one-dimensional spectra do not depend on the coupling. For bumps of bounded variation whose difference h = f_0 − f_1 is absolutely continuous with h′ of bounded variation we prove that the Fricke–Vogt invariant at energy t² is sin²t |h(1)e^{2it} − h(0)|²/(16t⁴) + O(t⁻⁵), with the same error bound for the t-derivative (Theorem C). If moreover h(0) + h(1) ≠ 0, this verifies the two hypotheses on the invariant in Proposition 4.2 of Damanik–Fillman–Gorodetski, one of which they name as the main obstacle for general bumps, and the half-line follows for all real couplings (Theorem B). Theorem B is conditional: it assumes the assertion of their Remark 4.25, not proved there, that their construction of the curve pieces and its itinerary property extend to general bumps. Bumps with h(0) + h(1) = 0, which include all continuous potentials V_ω, and all other bumps outside the hypotheses of Theorem B, for instance bumps whose difference is not absolutely continuous, remain open. This is an unrefereed note.",
  "result_type": "PARTIAL_RESULT",
  "categories": [
    "math.SP",
    "math-ph",
    "math.DS"
  ],
  "keywords": [
    "continuum Fibonacci Schrödinger operators",
    "separable operators",
    "half-line in the spectrum",
    "sums of Cantor sets",
    "Newhouse thickness",
    "trace map",
    "Fricke–Vogt invariant",
    "Bethe–Sommerfeld",
    "partial answer",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-15951-003",
    "math.SP",
    "math-ph",
    "math.DS",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-15951-003/",
  "pdf_url": "https://eulersolve.org/papers/owr-15951-003/paper.pdf?v=cd022829d1b9",
  "doi": "10.5281/zenodo.23451885",
  "zenodo_record_url": "https://zenodo.org/records/23451885",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "A partial answer. Theorem A (constant bumps, all real couplings) uses the published inputs of Damanik, Fillman and Gorodetski; Theorem C is unconditional; Theorem B is conditional on an assertion of their Remark 4.25 that is not proved in print. Bumps with f_0(0)+f_0(1) = f_1(0)+f_1(1), including all continuous potentials, remain open. No priority is claimed.",
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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."
}