Delocalization of quasimodes on the disk

Anantharaman, Nalini; Léautaud, Matthieu; Macià, Fabricio

doi:10.1016/j.crma.2015.10.016

Partial differential equations/Mathematical physics

Delocalization of quasimodes on the disk
[Délocalisation des quasimodes sur le disque]

Présenté par : the Editorial Board

Anantharaman, Nalini ¹ ; Léautaud, Matthieu ² ; Macià, Fabricio 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, Tome 354 (2016) no. 3, pp. 257-263.

Résumé
Abstract

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.

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.

Reçu le : 2015-04-15
Accepté le : 2015-10-26
Publié le : 2016-01-29

DOI : 10.1016/j.crma.2015.10.016

Affiliations des auteurs :

Anantharaman, Nalini ¹ ; Léautaud, Matthieu ² ; Macià, Fabricio ³

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Anantharaman, Nalini; Léautaud, Matthieu; Macià, Fabricio. Delocalization of quasimodes on the disk. Comptes Rendus. Mathématique, Tome 354 (2016) no. 3, pp. 257-263. doi : 10.1016/j.crma.2015.10.016. http://www.numdam.org/articles/10.1016/j.crma.2015.10.016/

Bibliographie
Cité par

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Cité par 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.