On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $

In this paper, we establish a convergence result for the operator splitting scheme \begin{document}$ Z_{\tau} $\end{document} introduced by Ignat [12], with initial data in \begin{document}$ H^1 $\end{document}, for the nonlinear Schrödinger equation:

\begin{document}$ \partial_t u = i \Delta u + i\lambda |u|^{p} u,\qquad u (x,0) = \phi (x), $\end{document}

where \begin{document}$ p >0 $\end{document}, \begin{document}$ \lambda \in \{-1,1\} $\end{document} and \begin{document}$ (x,t) \in \mathbb{R}^d \times [0,\infty) $\end{document}. We prove the \begin{document}$ L^2 $\end{document} convergence of order \begin{document}$ \mathcal{O}(\tau^{1/2}) $\end{document} for the scheme with initial data in the space \begin{document}$ H^1 (\mathbb{R}^d) $\end{document} for the energy-subcritical range of \begin{document}$ p $\end{document}.