Comptes Rendus
Partial Differential Equations/Numerical Analysis
Long time behavior of splitting methods applied to the linear Schrödinger equation
Comptes Rendus. Mathématique, Volume 344 (2007) no. 2, pp. 89-92.

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.

Nous considérons la semi-discrétisation en temps de l'équation de Schrödinger linéaire sur un tore de dimension un par une méthode de splitting. Sous une condition de non-résonance sur le pas de temps et sous l'hypothèse que le potentiel est petit et analytique, nous montrons la conservation des énergies associées aux valeurs propres doubles du Laplacien sur des temps exponentiellement longs et pour des modes asymptotiquement grands. Le résultat repose sur un théorème de forme normale dont la preuve utilise la théorie classique des perturbations, appliquée ici à un problème de dimension infinie.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.11.024

Guillaume Dujardin 1; Erwan Faou 1

1 INRIA Rennes, campus Beaulieu, 35042 Rennes cedex, France
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     title = {Long time behavior of splitting methods applied to the linear {Schr\"odinger} equation},
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Guillaume Dujardin; Erwan Faou. Long time behavior of splitting methods applied to the linear Schrödinger equation. Comptes Rendus. Mathématique, Volume 344 (2007) no. 2, pp. 89-92. doi : 10.1016/j.crma.2006.11.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.11.024/

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