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In this paper, we establish a convergence result for the operator splittingscheme $Z_{\tau}$ introduced by Ignat, with initial data in $H^1$, for thenonlinear Schr\"odinger equation :  $$  \partial_t u = i \Delta u + i\lambda |u|^{p} u,\qquad u (x,0) =\phi (x),  $$ where $(x,t) \in \mathbb{R}^d \times [0,\infty)$, with $0< p < 4/(d-2)$for $d\geq3$ and $0< p<\infty$ for $d=1,2$. We prove the $L^2$ convergence oforder $\mathcal{O}(\tau^{1/2})$ for this scheme with initial data in the space$H^1 (\mathbb{R}^d)$.

On the splitting method for the nonlinear Schr\"odinger equation with initial data in $H^1$