Dans cette note, on s'intéresse aux mesures semiclassiques associées aux quasimodes (d'ordre suffisamment élevé) 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 : 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 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 : these are absolutely continuous in the interior of the disk and charge every open set intersecting the boundary.
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@article{CRMATH_2016__354_3_257_0, author = {Anantharaman, Nalini and L\'eautaud, Matthieu and Maci\`a, Fabricio}, title = {Delocalization of quasimodes on the disk}, journal = {Comptes Rendus. Math\'ematique}, pages = {257--263}, publisher = {Elsevier}, volume = {354}, number = {3}, year = {2016}, doi = {10.1016/j.crma.2015.10.016}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2015.10.016/} }
TY - JOUR AU - Anantharaman, Nalini AU - Léautaud, Matthieu AU - Macià, Fabricio TI - Delocalization of quasimodes on the disk JO - Comptes Rendus. Mathématique PY - 2016 SP - 257 EP - 263 VL - 354 IS - 3 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2015.10.016/ DO - 10.1016/j.crma.2015.10.016 LA - en ID - CRMATH_2016__354_3_257_0 ER -
%0 Journal Article %A Anantharaman, Nalini %A Léautaud, Matthieu %A Macià, Fabricio %T Delocalization of quasimodes on the disk %J Comptes Rendus. Mathématique %D 2016 %P 257-263 %V 354 %N 3 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2015.10.016/ %R 10.1016/j.crma.2015.10.016 %G en %F CRMATH_2016__354_3_257_0
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://archive.numdam.org/articles/10.1016/j.crma.2015.10.016/
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☆ 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.