Propagation des singularités et résonances

Jean-François Bony; Setsuro Fujiié; Thierry Ramond; Maher Zerzeri

doi:10.1016/j.crma.2017.06.008

Équations aux dérivées partielles/Physique mathématique

Propagation des singularités et résonances
[Propagation of singularities and resonances]

Presented by: le comité de rédaction

Jean-François Bony ¹; Setsuro Fujiié ²; Thierry Ramond ³; Maher Zerzeri ⁴

¹ IMB, CNRS (UMR 5251), université de Bordeaux, 33405 Talence, France
² Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Noji-Higashi, Kusatsu, 525-8577 Japan
³ Laboratoire de mathématiques d'Orsay, université Paris-Sud, CNRS, université Paris-Saclay, 91405 Orsay, France
⁴ Université Paris-13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 93430 Villetaneuse, France

Comptes Rendus. Mathématique, Volume 355 (2017) no. 8, pp. 887-891.

Abstract
Résumé

In the framework of semiclassical resonances, we make more precise the link between the polynomial estimates of the extension of the resolvent and the propagation of the singularities through the trapped set. This approach makes it possible to eliminate infinity and to concentrate the study near the trapped set. It has allowed us in previous papers to obtain the asymptotic of resonances in various geometric situations.

Dans le cadre de l'étude des résonances semiclassiques, on précise le lien entre majoration polynomiale du prolongement de la résolvante et propagation des singularités à travers l'ensemble capté. Cette approche permet d'éliminer l'infini et de concentrer l'étude près de l'ensemble capté. Nous l'avons utilisée dans des travaux antérieurs pour obtenir l'asymptotique des résonances dans diverses situations géométriques.

Received: 2017-04-13
Accepted: 2017-06-26
Published online: 2017-07-10

DOI: 10.1016/j.crma.2017.06.008

Author's affiliations:

Jean-François Bony ¹; Setsuro Fujiié ²; Thierry Ramond ³; Maher Zerzeri ⁴

¹ IMB, CNRS (UMR 5251), université de Bordeaux, 33405 Talence, France
² Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Noji-Higashi, Kusatsu, 525-8577 Japan
³ Laboratoire de mathématiques d'Orsay, université Paris-Sud, CNRS, université Paris-Saclay, 91405 Orsay, France
⁴ Université Paris-13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 93430 Villetaneuse, France

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     author = {Jean-Fran\c{c}ois Bony and Setsuro Fujii\'e and Thierry Ramond and Maher Zerzeri},
     title = {Propagation des singularit\'es et r\'esonances},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {887--891},
     publisher = {Elsevier},
     volume = {355},
     number = {8},
     year = {2017},
     doi = {10.1016/j.crma.2017.06.008},
     language = {fr},
}

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Jean-François Bony; Setsuro Fujiié; Thierry Ramond; Maher Zerzeri. Propagation des singularités et résonances. Comptes Rendus. Mathématique, Volume 355 (2017) no. 8, pp. 887-891. doi : 10.1016/j.crma.2017.06.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.06.008/

References
Cited by

[1] J.-F. Bony; S. Fujiié; T. Ramond; M. Zerzeri Barrier-top resonances for non globally analytic potentials, 2016 (preprint) | arXiv

[2] J.-F. Bony; S. Fujiié; T. Ramond; M. Zerzeri Resonances for homoclinic trapped sets, 2016 (preprint) | arXiv

[3] J.-F. Bony; V. Petkov Semiclassical estimates of the cut-off resolvent for trapping perturbations, J. Spectr. Theory, Volume 3 (2013) no. 3, pp. 399-422

[4] M. Dimassi; J. Sjöstrand Spectral Asymptotics in the Semi-Classical Limit, Lond. Math. Soc. Lect. Note Ser., vol. 268, Cambridge University Press, 1999

[5] S. Fujiié; A. Lahmar-Benbernou; A. Martinez Width of shape resonances for non globally analytic potentials, J. Math. Soc. Jpn., Volume 63 (2011) no. 1, pp. 1-78

[6] C. Gérard; J. Sjöstrand Semiclassical resonances generated by a closed trajectory of hyperbolic type, Commun. Math. Phys., Volume 108 (1987), pp. 391-421

[7] B. Helffer; J. Sjöstrand Résonances en limite semi-classique, Mém. Soc. Math. Fr. (1986) no. 24–25 (iv+228 p)

[8] A. Martinez An Introduction to Semiclassical and Microlocal Analysis, Universitext, Springer, 2002

[9] A. Martinez Resonance free domains for non globally analytic potentials, Ann. Henri Poincaré, Volume 3 (2002) no. 4, pp. 739-756

[10] S. Nakamura Scattering theory for the shape resonance model. I. Nonresonant energies, Ann. Inst. Henri Poincaré A, Phys. Théor., Volume 50 (1989) no. 2, pp. 115-131

[11] S. Nakamura Scattering theory for the shape resonance model. II. Resonance scattering, Ann. Inst. Henri Poincaré A, Phys. Théor., Volume 50 (1989) no. 2, pp. 133-142

[12] J. Sjöstrand Resonances for bottles and trace formulae, Math. Nachr., Volume 221 (2001), pp. 95-149

[13] P. Stefanov Resonance expansions and Rayleigh waves, Math. Res. Lett., Volume 8 (2001) no. 1–2, pp. 107-124

[14] S.-H. Tang; M. Zworski From quasimodes to resonances, Math. Res. Lett., Volume 5 (1998) no. 3, pp. 261-272

[15] M. Zworski Semiclassical Analysis, Grad. Stud. Math., vol. 138, American Mathematical Society, 2012

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