Sharp convergence for sequences of Schrödinger means and related generalizations

For decreasing sequences $\{t_n{\}}_{n=1}^{\mathrm{\infty }}$ converging to zero, we obtain the almost everywhere convergence results for sequences of Schrödinger means $e^{i{t}_n\mathrm{\Delta }}f$, where $f\in H^s({\mathbb{R}}^N),N\ge 2$. The convergence results are sharp up to the endpoints, and the method can also be applied to get the convergence results for the fractional Schrödinger means and nonelliptic Schrödinger means.