Time splitting method for nonlinear Schrödinger equation with rough initial data in $L^2$

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 Lie approximation $Z_L$ to the solution $u$ in the mass-subcritical range, $\max\left\{1,\frac{2}{d}\right\} \leq p < \frac{4}{d}$. Furthermore, our argument can be extended to the case of initial data $\phi\in H^s (\mathbb{R}^d)$ $(0<s\leq1)$, for which we obtain a convergence rate of order $\frac{s}{2-s}$ that breaks the natural order barrier $\frac{s}{2}$.