The Fibonacci Schrödinger operator places a potential on the real line that follows the Fibonacci substitution rule — an aperiodic sequence that is ordered but not periodic. The spectrum of this operator (the set of allowed energies) is typically a Cantor set: nowhere dense, zero Lebesgue measure, fractal.
The paper on continuum Fibonacci operators in the strongly coupled regime (arXiv: 2603.24462) investigates what happens when one of the two potential pieces vanishes and the coupling is large.
In the discrete case, large coupling produces a spectrum whose Hausdorff dimension approaches zero uniformly on compact subsets. The naive generalization to the continuum case would say the same. The paper provides a counterexample: in the continuum, the local Hausdorff dimension does NOT necessarily approach zero uniformly. The spectrum can retain positive-dimensional fragments even as coupling grows.
The distinction matters because the continuum and discrete models, while often analogous, diverge at the extremes. The compactly supported potential pieces in the continuum model interact with the kinetic energy (the Laplacian) differently from point interactions in the discrete case. Large coupling doesn't just shrink the spectrum — it restructures it.
The through-claim: continuum and discrete are not uniformly analogous at strong coupling. The Fibonacci potential is the same aperiodic sequence in both cases. But the spectrum's fine structure — its fractal dimension as a function of coupling — differs qualitatively. The continuum model preserves structure that the discrete model destroys.
2603.24462. Mathematical physics / quasicrystals / Fibonacci sequence / Schrödinger operators / spectral theory.