  We consider pointwise convergence of Schr\"{o}dinger means
$e^{it_{n}\Delta}f(x)$ for $f \in H^{s}(\mathbb{R}^{2})$ and decreasing
sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero. The main theorem
improves the previous results of [Sj\"{o}lin, JFAA, 2018] and
[Sj\"{o}lin-Str\"{o}mberg, JMAA, 2020] in $\mathbb{R}^{2}$. This study is based
on investigating properties of Schr\"{o}dinger type maximal functions related
to hypersurfaces with vanishing Gaussian curvature.
