Résonances Semiclassiques Engendrées par des Croisements de Trajectoires Classiques

Kenta Higuchi

doi:10.5802/crmath.209

Partial differential equations, Mathematical physics

Résonances Semiclassiques Engendrées par des Croisements de Trajectoires Classiques
[Semiclassical Resonances Generated by Crossings of Classical Trajectories]

Kenta Higuchi ¹

¹ Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Noji-Higashi, Kusatsu, 525-8577, Japan

Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 657-663.

Abstract
Résumé

We consider a $2 \times 2$ 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 $T^{- 1} h log (1 / h)$ , 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 $2 \times 2$ 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 $T^{- 1} h log (1 / h)$ , contrairement au cas scalaire, sous la condition que deux trajectoires classiques se croisent et composent une trajectoire périodique de période $T$ .

Received: 2021-03-05
Revised: 2021-04-02
Accepted: 2021-04-03
Published online: 2021-08-25

DOI: 10.5802/crmath.209

Classification: 35P15, 35C20, 35S99, 47A75

Author's affiliations:

Kenta Higuchi ¹

¹ Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Noji-Higashi, Kusatsu, 525-8577, Japan

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Kenta Higuchi},
     title = {R\'esonances {Semiclassiques} {Engendr\'ees} par des {Croisements} de {Trajectoires} {Classiques}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {657--663},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {6},
     year = {2021},
     doi = {10.5802/crmath.209},
     language = {fr},
}

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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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