Long time behavior of splitting methods applied to the linear Schrödinger equation

We consider the linear Schrödinger equation on a one-dimensional torus and its time-discretization by splitting methods. Assuming a non-resonance condition on the stepsize and a small analytical size of the potential, we show the conservation over exponentially long time of the energies associated with the double eigenvalues of the Laplace operator for asymptotically large modes. The result relies on a normal form theorem whose proof uses standard techniques of classical perturbations theory, extended here to an infinite dimensional context. To cite this article: G. Dujardin, E. Faou, C. R. Acad. Sci. Paris, Ser. I 344 (2007).