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On Fillman's Question for Separable Continuum Fibonacci Operators: Half-Lines for Two Couplings and the High-Energy Fricke–Vogt Invariant

Year: 2026
Type: preprint
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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-15951-003 (J. Fillman, Spectral properties of continuum quasicrystal models, Oberwolfach Report 46/2017, pp. 2797–2798).
DOI: 10.5281/zenodo.23451883
Language: English
Open Access status: green
OpenAlex ID: W7222888358
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