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.

Abstract (VO)
Résumé

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.

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.

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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     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

[1] Nalini Anantharaman Entropy and the localization of eigenfunctions, Ann. of Math. (2), Volume 168 (2008) no. 2, pp. 435-475

[2] Nalini Anantharaman, Herbert Koch, Stéphane Nonnenmacher, Entropy of eigenfunctions, preprint, 2007.

[3] Nalini Anantharaman; Stéphane Nonnenmacher Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Ann. Inst. Fourier (Grenoble), Volume 57 (2007) no. 7, pp. 2465-2523 (Festival Yves Colin de Verdière)

[4] Jean Bourgain; Elon Lindenstrauss Entropy of quantum limits, Comm. Math. Phys., Volume 233 (2003) no. 1, pp. 153-171

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

[6] Manfred Einsiedler; Anatole Katok; Elon Lindenstrauss Invariant measures and the set of exceptions to Littlewood's conjecture, Ann. of Math. (2), Volume 164 (2006) no. 2, pp. 513-560

[7] Roman Holowinsky, Kannan Soundararajan, Mass equidistribution of Hecke eigenforms, Ann. of Math., in press.

[8] Elon Lindenstrauss On quantum unique ergodicity for $Γ \ H \times H$ , Internat. Math. Res. Notices (17) (2001), pp. 913-933

[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

[11] Lior Silberman, Akshay Venkatesh, Entropy bounds for Hecke eigenfunctions on division algebras, GAFA (2010), in press.

[12] K. Soundararajan, Quantum unique ergodicity for ${SL}_{2} (Z) \ H$ , preprint, 2009.

[13] Scott A. Wolpert Semiclassical limits for the hyperbolic plane, Duke Math. J., Volume 108 (2001) no. 3, pp. 449-509

Paul D. Nelson Microlocal lifts and quantum unique ergodicity on ${GL}_{2} (Q_{p})$ , Algebra Number Theory, Volume 12 (2018) no. 9, pp. 2033-2064 | DOI:10.2140/ant.2018.12.2033 | Zbl:1406.58019
G. Berkolaiko; B. Winn Maximal scarring for eigenfunctions of quantum graphs, Nonlinearity, Volume 31 (2018) no. 10, pp. 4812-4850 | DOI:10.1088/1361-6544/aad3fe | Zbl:1397.81079
Shimon Brooks; Elon Lindenstrauss Non-localization of eigenfunctions on large regular graphs, Israel Journal of Mathematics, Volume 193 (2013), pp. 1-14 | DOI:10.1007/s11856-012-0096-y | Zbl:1317.05110
Jean-Philippe Anker; Pierre Martinot; Emmanuel Pedon; Alberto G. Setti The shifted wave equation on Damek-Ricci spaces and on homogeneous trees, Trends in harmonic analysis. Selected papers of the conference on harmonic analysis, Rome, Italy, May 30–June 4, 2011, Berlin: Springer, 2013, pp. 1-25 | DOI:10.1007/978-88-470-2853-1_1 | Zbl:1275.43010

Cité par 4 documents. Sources : zbMATH

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