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A typical procedure to integrate numerically the time dependentSchr\"o\-din\-ger equation involves two stages. In the first one carries out aspace discretization of the continuous problem. This results in the linearsystem of differential equations $i du/dt = H u$, where $H$ is a real symmetricmatrix, whose solution with initial value $u(0) = u_0 \in \mathbb{C}^N$ isgiven by $u(t) = \e^{-i t H} u_0$. Usually, this exponential matrix isexpensive to evaluate, so that time stepping methods to constructapproximations to $u$ from time $t_n$ to $t_{n+1}$ are considered in the secondphase of the procedure. Among them, schemes involving multiplications of thematrix $H$ with vectors, such as Lanczos and Chebyshev methods, areparticularly efficient.  In this work we consider a particular class of splitting methods which alsoinvolves only products $Hu$. We carry out an error analysis of theseintegrators and propose a strategy which allows us to construct differentsplitting symplectic methods of different order (even of order zero) possessinga large stability interval that can be adapted to different space regularityconditions and different accuracy ranges of the spatial discretization. Thevalidity of the procedure and the performance of the resulting schemes areillustrated on several numerical examples.

Error analysis of splitting methods for the time dependent Schrodinger equation