Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential

Abstract

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 size of the potential, we show that the numerical dynamics can be reduced over exponentially long time to a collection of two dimensional symplectic systems for asymptotically large modes. For the numerical solution, this implies the long time conservation of the energies associated with the double eigenvalues of the free Schrödinger operator. The method is close to standard techniques used in finite dimensional perturbation theory, but extended here to infinite dimensional operators.

Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations

“…Long-time conservation properties are obtained under a list of unverified conditions formulated as conjectures. For symplectic splitting methods applied to the linear Schrödinger equation with a small potential, results on long-time energy conservation are given by Dujardin & Faou [8].…”

Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation

“…The typical regularity of the potential function and of the unknown wave function is again of Gevrey type. To be slightly more precise, the results of this paper extend those presented in [6] in two ways: firstly, [6] only considers analytical solutions; secondly, no space discretization is made in [6] where only time discretizations are studied.…”

“…The aim of this paper is to extend the above result of [6] (see also [5,7]) in the fully discretized case, i.e. when space is sampled as well.…”

“…In that direction, it has been proved in [6] (see also [5,7]) that the building block e −i t n [−Δ] e −i t n λV (x) preserves regularity of u 0 , provided λ is small enough, and V is smooth. We definitely refer here to Gevrey regularity.…”

“…Our main theorem in this section is Theorem 3.13. Note that the techniques and the method of proof that we use here is directly inspired from [6] (see also [5,7]). …”