Graph eigenfunctions and quantum unique ergodicity

Shimon Brooks; Elon Lindenstrauss

doi:10.1016/j.crma.2010.07.003

Number Theory/Dynamical Systems

Graph eigenfunctions and quantum unique ergodicity
[Fonctions propres de graphes et l'unique ergodicité quantique]

Présenté par : Jean Bourgain

Shimon Brooks ¹ ; Elon Lindenstrauss ²

¹ Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY 11794, USA
² Einstein Institute of Mathematics, Givaat Ram, 91904 Jerusalem, Israel

Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 829-834.

Résumé
Abstract

On applique les techniques de Brooks et Lindenstrauss (2010) [5] pour étudier fonctions propres jointes du laplacien et d'un opérateur Hecke sur des surfaces compactes de congruence, et les fonctions propres jointes de deux laplaciens partiels sur les quotients compacts de $H \times H$ . Dans les deux cas, on montre entropie strictement positive sur presque toutes les composantes ergodiques des limites quantiques. De plus, les travaux de Lindenstrauss (2006) [9] ce implique Unique Ergodicité Quantique pour ces fonctions.

We apply the techniques of Brooks and Lindenstrauss (2010) [5] to study joint eigenfunctions of the Laplacian and one Hecke operator on compact congruence surfaces, and joint eigenfunctions of the two partial Laplacians on compact quotients of $H \times H$ . In both cases, we show that quantum limit measures of such sequences of eigenfunctions carry positive entropy on almost every ergodic component. Together with the work of Lindenstrauss (2006) [9], this implies Quantum Unique Ergodicity for such functions.

Reçu le : 2010-06-23
Accepté le : 2010-07-02
Publié le : 2010-08-01

DOI : 10.1016/j.crma.2010.07.003

Affiliations des auteurs :

Shimon Brooks ¹ ; Elon Lindenstrauss ²

¹ Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY 11794, USA
² Einstein Institute of Mathematics, Givaat Ram, 91904 Jerusalem, Israel

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     author = {Shimon Brooks and Elon Lindenstrauss},
     title = {Graph eigenfunctions and quantum unique ergodicity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {829--834},
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     volume = {348},
     number = {15-16},
     year = {2010},
     doi = {10.1016/j.crma.2010.07.003},
     language = {en},
}

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Shimon Brooks; Elon Lindenstrauss. Graph eigenfunctions and quantum unique ergodicity. Comptes Rendus. Mathématique, Volume 348 (2010) no. 15-16, pp. 829-834. doi : 10.1016/j.crma.2010.07.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.003/

Bibliographie
Cité par

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[5] Shimon Brooks, Elon Lindenstrauss, Non-localization of eigenfunctions on large regular graphs, 2010, submitted for publication.

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[9] Elon Lindenstrauss Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2), Volume 163 (2006) no. 1, pp. 165-219

[10] Lior Silberman; Akshay Venkatesh On quantum unique ergodicity for locally symmetric spaces, Geom. Funct. Anal., Volume 17 (2007) no. 3, pp. 960-998

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