  A typical procedure to integrate numerically the time dependent
Schr\"o\-din\-ger equation involves two stages. In the first one carries out a
space discretization of the continuous problem. This results in the linear
system of differential equations $i du/dt = H u$, where $H$ is a real symmetric
matrix, whose solution with initial value $u(0) = u_0 \in \mathbb{C}^N$ is
given by $u(t) = \e^{-i t H} u_0$. Usually, this exponential matrix is
expensive to evaluate, so that time stepping methods to construct
approximations to $u$ from time $t_n$ to $t_{n+1}$ are considered in the second
phase of the procedure. Among them, schemes involving multiplications of the
matrix $H$ with vectors, such as Lanczos and Chebyshev methods, are
particularly efficient.
  In this work we consider a particular class of splitting methods which also
involves only products $Hu$. We carry out an error analysis of these
integrators and propose a strategy which allows us to construct different
splitting symplectic methods of different order (even of order zero) possessing
a large stability interval that can be adapted to different space regularity
conditions and different accuracy ranges of the spatial discretization. The
validity of the procedure and the performance of the resulting schemes are
illustrated on several numerical examples.
