Effective stability for Hamiltonian PDEs vanishing spectral gaps

This paper studies the effective stability of the space fractional Schr\"odinger equation for solutions with various regularities. The study focuses on the regime $0 < \beta < 1/2$, a regime characterized by asymptotic frequencies. We utilize a high-low frequency decomposition to overcome the challenge of these asymptotic frequencies, showing that the errors arising from near-resonances can be controlled and effectively absorbed by the inherent smallness of the solution's high-frequency part. Furthermore, this framework uniformly yields stability time estimates for solutions in the Gevrey class, logarithmic ultra-differentiable, and finitely differentiable spaces.