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Dynamics near the origin of the long range scattering for the one-dimensional Schrödinger equation
[Dynamique près de l’origine du scattering longue portée pour l’équation de Schrödinger unidimensionnelle]
Rémi Carles1
1 Univ Rennes, CNRS, IRMAR – UMR 6625, F-35000 Rennes, France
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1717-1742.

Nous considèrons l’équation de Schrödinger cubique sur la droite, pour laquelle la théorie du scattering demande des modifications dues aux effets à longue portée. Nous reprenons la construction de l’opérateur d’onde modifié, et rappelons la construction de son inverse, afin de décrire le comportement de ces opérateurs près de l’origine. Au premier ordre, ces opérateurs, dont la définition contient une modification non linéaire de la phase par rapport à la dynamique linéaire, coïncident avec l’identité. Nous calculons explicitement le premier correcteur du développement asymptotique, et justifions ce développement par des estimations d’erreur.

We consider the cubic Schrödinger equation on the line, for which the scattering theory requires modifications due to long range effects. We revisit the construction of the modified wave operator, and recall the construction of its inverse, in order to describe the asymptotic behavior of these operators near the origin. At leading order, these operators, whose definition includes a nonlinear modification in the phase compared to the linear dynamics, correspond to the identity. We compute explicitly the first corrector in the asymptotic expansion, and justify this expansion by error estimates.

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Révisé le :
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DOI : 10.5802/crmath.676
Classification : 35Q55, 35P25, 35B40
Keywords: Nonlinear Schrödinger equation, long range scattering, asymptotic expansion, error estimate
Mots-clés : Équation de Schrödinger non linéaire, scattering longue portée, développement asymptotique, estimation d’erreur

Rémi Carles 1

1 Univ Rennes, CNRS, IRMAR – UMR 6625, F-35000 Rennes, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Dynamics near the origin of the long range scattering for the one-dimensional {Schr\"odinger} equation},
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     year = {2024},
     doi = {10.5802/crmath.676},
     language = {en},
}
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Rémi Carles. Dynamics near the origin of the long range scattering for the one-dimensional Schrödinger equation. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1717-1742. doi : 10.5802/crmath.676. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.676/

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