Time splitting method for nonlinear Schr\"odinger 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 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 $L^2$ 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 $\phi\in L^2 (\mathbb{R}^d)$.