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 (Original language)
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},
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     pages = {257--263},
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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
Cited by

[1] Nalini Anantharaman; Clotilde Fermanian-Kammerer; Fabricio Macià Semiclassical completely integrable systems: long-time dynamics and observability via two-microlocal Wigner measures, Amer. J. Math., Volume 137 (2015) no. 3, pp. 577-638

[2] Nalini Anantharaman; Matthieu Léautaud Sharp polynomial decay rates for the damped wave equation on the torus, Anal. PDE, Volume 7 (2014) no. 1, pp. 159-214

[3] Nalini Anantharaman; Matthieu Léautaud; Fabricio Macià Wigner measures and observability for the Schrödinger equation on the disk, 2014 (submitted for publication) | arXiv

[4] Nalini Anantharaman; Fabricio Macià Semiclassical measures for the Schrödinger equation on the torus, J. Eur. Math. Soc., Volume 16 (2014) no. 6, pp. 1253-1288

[5] Nicolas Burq; Maciej Zworski Geometric control in the presence of a black box, J. Amer. Math. Soc., Volume 17 (2004) no. 2, pp. 443-471 (electronic)

[6] Nicolas Burq; Maciej Zworski Control for Schrödinger operators on tori, Math. Res. Lett., Volume 19 (2012) no. 2, pp. 309-324

[7] Patrick Gérard Microlocal defect measures, Commun. Partial Differ. Equ., Volume 16 (1991) no. 11, pp. 1761-1794

[8] Patrick Gérard; Éric Leichtnam Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J., Volume 71 (1993) no. 2, pp. 559-607

[9] Stéphane Jaffard Contrôle interne exact des vibrations d'une plaque rectangulaire, Port. Math., Volume 47 (1990) no. 4, pp. 423-429

[10] Vilmos Komornik On the exact internal controllability of a Petrowsky system, J. Math. Pures Appl. (9), Volume 71 (1992) no. 4, pp. 331-342

[11] Gilles Lebeau Contrôle de l'équation de Schrödinger, J. Math. Pures Appl. (9), Volume 71 (1992) no. 3, pp. 267-291

[12] Gilles Lebeau; Luc Robbiano Contrôle exact de l'équation de la chaleur, Commun. Partial Differ. Equ., Volume 20 (1995) no. 1–2, pp. 335-356

[13] Fabricio Macià High-frequency propagation for the Schrödinger equation on the torus, J. Funct. Anal., Volume 258 (2010) no. 3, pp. 933-955

[14] Fabricio Macià The Schrödinger flow in a compact manifold: high-frequency dynamics and dispersion, Modern Aspects of the Theory of Partial Differential Equations, Oper. Theory, Adv. Appl., vol. 216, Birkhäuser/Springer Basel AG, Basel, 2011, pp. 275-289

[15] Miller Luc Controllability cost of conservative systems: resolvent condition and transmutation, J. Funct. Anal., Volume 218 (2005) no. 2, pp. 425-444

[16] Karim Ramdani; Takéo Takahashi; Gérald Tenenbaum; Marius Tucsnak A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator, J. Funct. Anal., Volume 226 (2005) no. 1, pp. 193-229

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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