Half-Line Spectra for Fibonacci Schrödinger Operators

Definitions and mathematical statement

Let $V_\omega$ denote the continuum Fibonacci potential on ${\mathbb{R}}$ referred to in equations (1)--(2) of the supplied Oberwolfach source. Those equations are absent from the supplied record, so the potential's construction, its parameters, and the admissible values of $\omega$ cannot be reconstructed here without additional source information.

For positive real couplings $\lambda_1,\lambda_2$, the source considers the separable Schrödinger operator on $L^2({\mathbb{R}}^2)$ with differential expression \[ L^{(2)}_{\lambda_1V_\omega,\lambda_2V_\omega} =-\Delta+\lambda_1V_\omega(x)+\lambda_2V_\omega(y), \qquad \Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}. \] The same potential $V_\omega$ occurs in both coordinate directions. The spatial domain is the whole plane; the supplied record specifies neither an operator domain nor the hypotheses defining the intended self-adjoint realization. Write $\sigma(L)$ for the spectrum of that intended realization.

The question is whether, for the source's potential $V_\omega$, \[ \forall\lambda_1,\lambda_2\in(0,\infty)\quad \exists E_0\in{\mathbb{R}}\quad [E_0,\infty)\subseteq \sigma\!\left(L^{(2)}_{\lambda_1V_\omega,\lambda_2V_\omega}\right). \] Here the threshold $E_0$ may depend on the two couplings and the potential data. The record does not specify the quantifier over $\omega$ or over any parameters in the missing potential definition. Thus the displayed question preserves the stated coupling quantifiers, but its full scope and operator realization require recovery of the referenced definitions.

Short English statement

For the continuum Fibonacci potential specified by the source, does the spectrum of the associated separable Schrödinger operator on ${\mathbb{R}}^2$ contain every energy above some threshold for every pair of positive couplings? The potential's defining equations, the scope of $\omega$, and the operator domain are missing from the supplied record.

Sources

• Local UnsolvedMath registry, supplied source record with permanent ID $30003616$, title: Half-Line Spectra for Fibonacci Schrödinger Operators; exact source problem code: catalogue entry. The record gives proposed year 2017 and names no proposer.
• Supplied source citation: Spectral Structures and Topological Methods in Mathematical Quasicrystals (2018), Oberwolfach Reports, pp. 2781--2845, DOI 10.4171/owr/2017/46. https://doi.org/10.4171/owr/2017/46. The literature-source label is Oberwolfach Report 46 2017. The original statement refers to equations (1)--(2), which are not included in the supplied record.
• Supplied literature reference, labelled Damanik Fillman Gorodetski 2020: https://doi.org/10.1016/j.jfa.2020.108911. No paper title or theorem-level attribution is supplied.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open and reports high-energy half-line results for important Fibonacci operators and some parameter regimes, while retaining the question for every pair of positive couplings. This is supplied dated curation, not independently verified here; the record requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a check against the official Oberwolfach report. This is a supplied verification claim, not an independent source check performed for this draft.