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 Schrödinger equation with rough initial data in $L^2$, $$\{\begin{array}{lll}i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u=\lambda \mathrm{\mid }u{\mathrm{\mid }}^{p}u,& (x,t)\in {\mathbb{R}}^d\times {\mathbb{R}}_{+},u(x,0)=\phi (x),& x\in {\mathbb{R}}^d,\end{array}$$ 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 $L^2$ convergence of the filtered Lie approximation $Z_{flt}$ to the solution $u$ in the mass-subcritical range, $0<p<\frac{4}{d}$. Furthermore, we provide a precise convergence result for radial initial data $\phi \in L^2({\mathbb{R}}^d)$.