Singular continuous spectrum for regular Floquet functions

Determine whether crystals whose Floquet matrix entries h_{ij}(θ) are regular (e.g., smooth or Lipschitz, beyond analyticity) can exhibit singular continuous spectrum for their associated operator H_Γ.

Background

The paper gives an explicit example of a crystal with purely singular continuous spectrum constructed from a highly non-differentiable Floquet function, and notes that analyticity precludes singular continuous spectrum via standard theory.

This motivates asking whether intermediate regularity classes (smooth/Lipschitz but not analytic) can still allow singular continuous spectrum in this periodic, non-locally finite setting.