We establish convergence results related to the operator splitting scheme on the Cauchy problem for the nonlinear Schr\"odinger equation with rough initial data in $L^2$, $$ \left\{ \begin{array}{ll} i\partial_t u +\Delta u = \lambda |u|^{p} u, & (x,t) \in \mathbb{R}^d \times \mathbb{R}_+, u (x,0) =\phi (x), & x\in\mathbb{R}^d, \end{array} \right. $$ where $\lambda \in \{-1,1\}$ and $p >0$. While the Lie approximation $Z_L$ is known to converge to the solution $u$ when the initial datum $\phi$ is sufficiently smooth, the convergence result for rough initial data is open to question. In this paper, for rough initial data $\phi\in L^2 (\mathbb{R}^d)$, we prove the convergence of the filtered Lie approximation $Z_{flt}$ to the solution $u$ in the mass-subcritical range, $\max\left\{1,\frac{2}{d}\right\} \leq p < \frac{4}{d}$. Furthermore, we provide a precise convergence result for radial initial data $\phi\in L^2 (\mathbb{R}^d)$,

翻译：本文研究了非线性薛定谔方程在$L^2$粗糙初值下Cauchy问题的算子分裂格式收敛性，其中方程形式为：$$ \left\{ \begin{array}{ll} i\partial_t u +\Delta u = \lambda |u|^{p} u, & (x,t) \in \mathbb{R}^d \times \mathbb{R}_+, \\ u (x,0) =\phi (x), & x\in\mathbb{R}^d, \end{array} \right. $$ 其中$\lambda \in \{-1,1\}$且$p >0$。已知当初始数据$\phi$足够光滑时，Lie近似$Z_L$收敛到解$u$，但粗糙初值下的收敛性问题仍未解决。本文针对粗糙初值$\phi\in L^2 (\mathbb{R}^d)$，证明了在质量次临界范围$\max\left\{1,\frac{2}{d}\right\} \leq p < \frac{4}{d}$内，滤波Lie近似$Z_{flt}$收敛到解$u$。此外，我们给出了径向初值$\phi\in L^2 (\mathbb{R}^d)$的精确收敛结果。