Sharp convergence for sequences of nonelliptic Schr\"{o}dinger means

We consider pointwise convergence of nonelliptic Schr\"{o}dinger means $e^{it_{n}\square}f(x)$ for $f \in H^{s}(\mathbb{R}^{2})$ and decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero, where $${e^{it_{n}\square }}f\left( x \right): = \int_{{\mathbb{R}^2}} {{e^{i\left( {x \cdot \xi + t_{n}{{ \xi_{1}\xi_{2} }}} \right)}}\widehat{f}} \left( \xi \right)d\xi .$$ We prove that when $0<s < \frac{1}{2}$, $$\mathop {\lim }\limits_{n \to \infty} {e^{it_{n}\square }}f\left( x \right) = f(x) \hspace{0.2cm} a.e.\hspace{0.2cm} x\in \mathbb{R}^2$$ holds for all $f \in {H^s}\left( {{\mathbb{R}^2}} \right)$ if and only if $\{t_{n}\}_{n=1}^{\infty} \in \ell^{r(s), \infty}(\mathbb{N})$, $r(s)=\frac{s}{1-s}$. Moreover, our result remains valid in general dimensions.