Delocalization of quasimodes on the disk

Nalini Anantharaman; Matthieu Léautaud; Fabricio Macià

doi:10.1016/j.crma.2015.10.016

Partial differential equations/Mathematical physics

Delocalization of quasimodes on the disk

Presented by: the Editorial Board

Nalini Anantharaman ¹; Matthieu Léautaud ²; Fabricio Macià ³

¹ Université de Strasbourg, IRMA, 7, rue René-Descartes, 67084 Strasbourg cedex, France
² Université Paris-Diderot, Institut de mathématiques de Jussieu–Paris Rive gauche, UMR 7586, bâtiment Sophie-Germain, 75205 Paris cedex 13, France
³ Universidad Politécnica de Madrid, DCAIN, ETSI Navales, Avda. Arco de la Victoria s/n, 28040 Madrid, Spain

Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 257-263.

Abstract
Résumé

This note deals with semiclassical measures associated with (sufficiently accurate) quasimodes $(u_{h})$ for the Laplace–Dirichlet operator on the disk. In this time-independent set-up, we simplify the statements of [3] and their proofs. We describe the restriction of semiclassical measures to every invariant torus in terms of two-microlocal measures. As corollaries, we show regularity and delocalization properties for limit measures of $| u_{h} |^{2} d x$ : these are absolutely continuous in the interior of the disk and charge every open set intersecting the boundary.

Dans cette note, on s'intéresse aux mesures semiclassiques associées aux quasimodes (d'ordre suffisamment élevé) $(u_{h})$ du laplacien de Dirichlet sur le disque. Dans ce contexte stationnaire, les résultats obtenus dans [3] et leurs preuves sont simplifiés. On décrit la restriction de ces mesures à chaque tore invariant au moyen de mesures deux-microlocales. En corollaire, on montre des propriétés de régularité et de délocalisation des mesures limites des $| u_{h} |^{2} d x$ : celles-ci sont absolument continues à l'intérieur du disque et chargent tout ouvert qui touche le bord.

Received: 2015-04-15
Accepted: 2015-10-26
Published online: 2016-01-29

DOI: 10.1016/j.crma.2015.10.016

Author's affiliations:

Nalini Anantharaman ¹; Matthieu Léautaud ²; Fabricio Macià ³

¹ Université de Strasbourg, IRMA, 7, rue René-Descartes, 67084 Strasbourg cedex, France
² Université Paris-Diderot, Institut de mathématiques de Jussieu–Paris Rive gauche, UMR 7586, bâtiment Sophie-Germain, 75205 Paris cedex 13, France
³ Universidad Politécnica de Madrid, DCAIN, ETSI Navales, Avda. Arco de la Victoria s/n, 28040 Madrid, Spain

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     author = {Nalini Anantharaman and Matthieu L\'eautaud and Fabricio Maci\`a},
     title = {Delocalization of quasimodes on the disk},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {257--263},
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     year = {2016},
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     language = {en},
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Nalini Anantharaman; Matthieu Léautaud; Fabricio Macià. Delocalization of quasimodes on the disk. Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 257-263. doi : 10.1016/j.crma.2015.10.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.10.016/

References
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Cited by Sources:

^☆ NA and ML are partially supported by the Agence Nationale de la Recherche under grant GERASIC ANR-13-BS01-0007-01. FM is partially supported by grants MTM2013-41780-P (MEC) and ERC Starting Grant 277778.

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