Scattering resonances for highly oscillatory potentials

Drouot, Alexis

doi:10.24033/asens.2368

Scattering resonances for highly oscillatory potentials
[Résonances de potentiels rapidement oscillants]

Drouot, Alexis

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 4, pp. 865-925.

Résumé
Abstract

Nous étudions les résonances de potentiels à support compact $V_{ε} (x) = W (x, x / ε)$ , où $W : ℝ^{d} \times ℝ^{d} / {(2 π ℤ)}^{d} \to ℂ$ et $d$ est impair. Ainsi, $V_{ε}$ est la somme d'un potentiel qui varie lentement $W_{0}$ et d'un potentiel qui oscille à fréquence $1 / ε$ . Quand $W_{0} \equiv 0$ nous prouvons que $V_{ε}$ n'a pas de résonances dans la zone ${ℑ λ \geq - A ln (ε^{- 1})}$ mise à part une unique résonance proche de 0 si $d = 1$ . Nous montrons par un exemple explicite que ce résultat est optimal. Cela prouve une conjecture de Duchêne-Vukićević-Weinstein [12]. Quand $W_{0} \neq 0$ et $W$ est lisse nous montrons que les resonances de $V_{ε}$ qui restent bornées lorsque $ε$ tend vers 0 admettent une expansion en puissances de $ε$ . Les arguments de la preuve permettent de calculer les coefficients de cette expansion. Nous construisons un potentiel effectif qui converge uniformément vers $W_{0}$ lorsque $ε$ tend vers 0 et dont les résonances sont à distance $O (ε^{4})$ de celles de $W_{0}$ . Cela améliore et étend les résultats de Duchêne, Vukićević et Weinstein à toutes les dimensions impaires.

We study resonances of compactly supported potentials $V_{ε} (x) = W (x, x / ε)$ where $W : ℝ^{d} \times ℝ^{d} / {(2 π ℤ)}^{d} \to ℂ$ , $d$ odd. That means that $V_{ε}$ is a sum of a slowly varying potential, $W_{0}$ , and one oscillating at frequency $1 / ε$ . When $W_{0} \equiv 0$ we prove that there are no resonances above the line $ℑ λ = - A ln (ε^{- 1})$ , except a simple resonance near 0 when $d = 1$ . We show that this result is optimal by constructing a one-dimensional example. This settles a conjecture of Duchêne-Vukićević-Weinstein [12]. When $W_{0} \neq 0$ and $W$ smooth we prove that resonances in fixed strips admit an expansion in powers of $ε$ . The argument provides a method for computing the coefficients of the expansion. We produce an effective potential converging uniformly to $W_{0}$ as $ε \to 0$ and whose resonances approach resonances of $V_{ε}$ modulo $O (ε^{4})$ . This improves the one-dimensional result of Duchêne, Vukićević and Weinstein and extends it to all odd dimensions.

Publié le : 2018-11-12

MR Zbl

DOI : 10.24033/asens.2368

Classification : 35P15, 35P25, 42B20.
Keywords: Scattering resonances, highly oscillatory potentials, asymptotic expansions.
Mot clés : Résonances, potentiels rapidement oscillants, expansions asymptotiques.

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     author = {Drouot, Alexis},
     title = {Scattering resonances for highly oscillatory potentials},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {865--925},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {4},
     year = {2018},
     doi = {10.24033/asens.2368},
     mrnumber = {3861565},
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     url = {http://www.numdam.org/articles/10.24033/asens.2368/}
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Drouot, Alexis. Scattering resonances for highly oscillatory potentials. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 4, pp. 865-925. doi : 10.24033/asens.2368. http://www.numdam.org/articles/10.24033/asens.2368/

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