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

J. Fillman asked (Oberwolfach Report 46/2017) whether, for the continuum Fibonacci potential \(V_\omega\) built from two bump functions \(f_0,f_1\in L^2[0,1)\) on unit cells, the spectrum of the separable operator \(-\Delta+\lambda_1V_\omega(x)+\lambda_2V_\omega(y)\) on \(L^2(\mathbb{R}^2)\), which is the sum \(\sigma(L_{\lambda_1V_\omega})+\sigma(L_{\lambda_2V_\omega})\) of two one-dimensional spectra, contains a half-line for all \(\lambda_1,\lambda_2>0\). We give a partial answer. For locally constant bumps \(f_j=c_j\chi_{[0,1)}\) the answer is yes for all real \(\lambda_1,\lambda_2\) (Theorem A). For \(\lambda_1=\lambda_2\) 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 \(t^2\) is \(\sin^2t\,|h(1)e^{2it}-h(0)|^2/(16t^4)+O(t^{-5})\), with the same error bound for the \(t\)-derivative (Theorem C). If moreover \(h(0)+h(1)\ne0\), 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 \(V_\omega\), 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.