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Astérisque - Titles - 390 (2017) 77-100

Titles < 2017 < 390

Séminaire Bourbaki, volume 2015/2016, exposés 1104-1119
Astérisque 390 (2017), xi+533 pages
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Presentation, Summary

Exposé 1107 : Bornes de Weyl fractales et résonances
Frédéric NAUD
Astérisque 390 (2017), 77-100
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Résumé :
Hermann Weyl a démontré en 1911 un théorème remarquable sur la répartition asymptotique des valeurs propres du laplacien pour les domaines compacts à bord dans l'espace euclidien. Dans le cas des domaines non compacts de volume infini, il existe une notion naturelle qui généralise celle de valeur propre: les résonances. Les résonances forment un ensemble discret de nombres complexes dont les parties réelles sont liées à une fréquence d'oscillation tandis que la partie imaginaire traduit un taux d'amortissement. Un travail récent de Nonnenmacher-Sjöstrand-Zworski établit des bornes supérieures sur la densité des résonances lorsqu'on les compte dans une bande horizontale du plan complexe. Le taux de croissance fait apparaître, contrairement à la loi de Weyl classique, un exposant non entier lié à la dimension de Minkowski des trajectoires captées : c'est ce qu'on appelle une borne de Weyl fractale. Nous ferons une introduction à la notion de résonance et mettrons en perspective le travail de N-S-Z en faisant un historique des résultats précédents de la théorie.

Mots-clefs : Laplacien, résonances, dynamique hyperbolique.

Abstract:
Exposé 1107 : Fractal Weyl bounds and resonances
In 1911, Hermann Weyl proved a celebrated theorem on the asymptotic distribution of the eigenvalues of the Euclidean Laplacian on a bounded domain (Weyl's law). On non-compact domains (or manifolds) there exists a natural replacement data for the missing eigenvalues of the Laplacian, which is called resonances or scattering poles. Resonances form a discrete set of complex numbers whose real parts are related to frequency while imaginary parts reflect a damping rate. A recent result of Nonnenmacher-Sjöstrand-Zworski is concerned with upper bound on the density of resonances when counting resonances in a fixed horizontal strip. The growth rate is a power law with a non-integer exponent related to the Minkowski dimension of trapped billiard trajectories inside the (non-compact) domain. This is an example of a ``fractal'' Weyl bound. To put the work of N-S-Z into perspective, we will first try to provide an ``elementary'' introduction to the notion of resonance and then review some of the classical results that led to the most recent developments.

Keywords: Laplace operator, resonances and hyperbolic dynamics.

Class. math. : 35P20, 35P25.


ISSN : 0303-1179
Publié avec le concours de : Centre National de la Recherche Scientifique

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