Absolute continuity throughout the supercritical regime

Determine whether the spectrum of the Weierstrass long-range crystal is purely absolutely continuous for every supercritical parameter pair satisfying $\\lambda>\\mu^{-1}$.

Background

The paper proves pure absolute continuity for sufficiently large λ\lambda relative to μ−1\mu^{-1}, but does not settle the entire supercritical region. The unresolved question is whether absolute continuity persists for all λ>μ−1\lambda>\mu^{-1}, including parameters close to criticality.

References

It is natural to ask more generally whether the spectrum is purely absolutely continuous throughout the supercritical regime $\lambda>\mu{-1}$.

— Arbitrarily Fast Quantum Dispersion in Long-Range Crystals  (2608.27326 - Leclerc et al., 27 Aug 2026) in Section 7, subsection “Spectrum”