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

Abstract

We establish convergence results related to the operator splitting scheme on the Cauchy problem for the nonlinear Schrödinger equation with rough initial data in $L^2$, $$\{\begin{array}{llc}i{\partial }_{t}u+\hat{I}”u=\hat{I}»|u{|}^{p}u,& (x,t)\in {\mathbb{R}}^d\times {\mathbb{R}}_{+},u(x,0)=\ddot{I}•(x),& x\in {\mathbb{R}}^d,\end{array}$$ where $\hat{I}»\in \{-1,1\}$ and $p>0$. While the Lie approximation $Z_L$ is known to converge to the solution $u$ when the initial datum $\ddot{I}•$ is sufficiently smooth, the convergence result for rough initial data is open to question. In this paper, for rough initial data $\ddot{I}•\in L^2({\mathbb{R}}^d)$, 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 $\ddot{I}•\in L^2({\mathbb{R}}^d)$.