Generic validity of the nondegeneracy condition on band phase functions and decay when it fails

Determine whether, for periodic Jacobi operators J on the discrete half-line N with minimal period q≥1, and for each spectral band I_j with phase function k_j:[−π,0]→I_j defined by Δ(k_j(φ))=2cos(φ), the nondegeneracy condition that k''_j(φ)=0 implies k'''_j(φ)≠0 holds always or generically; and, in cases where this condition fails, ascertain the correct ℓ^1→ℓ^∞ dispersive time-decay rate for the propagator e^{−itJ}P_c.

Background

The global ℓ^1→ℓ^∞ decay bound proved in the paper (t^{−1/3}) relies on a nondegeneracy hypothesis for the phase functions k_j that parametrize the continuous spectrum via the discriminant Δ. Specifically, the proof assumes that whenever k''_j(φ)=0, one has k'''_j(φ)≠0, enabling application of a van der Corput estimate with s=3. While this condition holds for the discrete Laplacian and is verified in certain explicit models (e.g., SSH), the authors do not establish it in general for all periodic Jacobi operators on N.

Clarifying whether this nondegeneracy is universal or generic would settle the robustness of the t^{−1/3} decay. If the condition fails, the current analysis does not determine the optimal global decay rate, motivating a precise characterization of the decay exponent under degeneracies of higher order in the phase.