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 Schr\"odinger equation with rough initial data in $L<sup>2$, $$ \left{ \begin{array}{ll} i\partial_t u +\Delta u = \lambda |u|^{p} u, &amp; (x,t) \in \mathbb{R}^d \times \mathbb{R}<em>+, u (x,0) =\phi (x), &amp; x\in\mathbb{R}^d, \end{array} \right. $$ where $\lambda \in {-1,1}$ and $p &gt;0$. 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 $\phi\in L² (\mathbb{R}<sup>d)$, we prove the $L<sup>2$ convergence of the filtered Lie approximation $Z</em>{flt}$ to the solution $u$ in the mass-subcritical range, $0&lt; p &lt; \frac{4}{d}$. Furthermore, we provide a precise convergence result for radial initial data $\phi\in L² (\mathbb{R}<sup>d)$.