We consider the linear Schr\"odinger equation and its discretization by
split-step methods where the part corresponding to the Laplace operator is
approximated by the midpoint rule. We show that the numerical solution
coincides with the exact solution of a modified partial differential equation
at each time step. This shows the existence of a modified energy preserved by
the numerical scheme. This energy is close to the exact energy if the numerical
solution is smooth. As a consequence, we give uniform regularity estimates for
the numerical solution over arbitrary long tim
