Pointwise convergence of sequential Schrödinger means

We study pointwise convergence of the fractional Schrödinger means along sequences $t_n$ which converge to zero. Our main result is that bounds on the maximal function ${\sup }_n\mathrm{\mid }{e}^{i{t}_n(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}}f\mathrm{\mid }$ can be deduced from those on ${\sup }_{0<t\le 1}\mathrm{\mid }{e}^{it(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}}f\mathrm{\mid }$ when $\{t_n\}$ is contained in the Lorentz space ${\mathrm{\ell }}^{r,\mathrm{\infty }}$. Consequently, our results provide seemingly optimal results in higher dimensions, which extend the recent work of Dimou-Seeger, and Li-Wang-Yan to higher dimensions. Our approach based on a localization argument also works for other dispersive equations and provides alternative proofs of previous results on sequential convergence.