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Comptes Rendus. Mathématique
Partial differential equations, Mathematical physics
Résonances Semiclassiques Engendrées par des Croisements de Trajectoires Classiques
[Semiclassical Resonances Generated by Crossings of Classical Trajectories]
Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 657-663.

We consider a 2×2 system of one-dimensional semiclassical Schrödinger operators with small interactions with respect to the semiclassical parameter h. We study the asymptotics in the semiclassical limit of the resonances near a non-trapping energy for both corresponding classical Hamiltonians. We show the existence of resonances of width T -1 hlog(1/h), contrary to the scalar case, under the condition that two classical trajectories cross and compose a periodic trajectory with period T.

Nous considérons un système 2×2 d’opérateurs de Schrödinger semiclassique 1D avec petites interactions par rapport au paramètre semiclassique h. Nous étudions l’asymptotique des résonances en limite semiclassique près d’une énergie non-captive pour les deux hamiltoniens classiques correspondants. Nous montrons l’existence de résonances de largeur T -1 hlog(1/h), contrairement au cas scalaire, sous la condition que deux trajectoires classiques se croisent et composent une trajectoire périodique de période T.

Received:
Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/crmath.209
Classification: 35P15,  35C20,  35S99,  47A75
Kenta Higuchi 1

1. Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Noji-Higashi, Kusatsu, 525-8577, Japan
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     title = {R\'esonances {Semiclassiques} {Engendr\'ees} par des {Croisements} de {Trajectoires} {Classiques}},
     journal = {Comptes Rendus. Math\'ematique},
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Kenta Higuchi. Résonances Semiclassiques Engendrées par des Croisements de Trajectoires Classiques. Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 657-663. doi : 10.5802/crmath.209. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.209/

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