Microlocal limits of plane waves  and Eisenstein functions

Dyatlov, Semyon; Guillarmou, Colin

doi:10.24033/asens.2217

Microlocal limits of plane waves and Eisenstein functions
[Limites microlocales des ondes planes et les fonctions d'Eisenstein]

Dyatlov, Semyon ; Guillarmou, Colin

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 2, pp. 371-448.

Résumé
Abstract

Dans ce travail, nous étudions les mesures microlocales des fonctions de type ondes planes sur des variétés non compactes $(M, g)$ qui, près de l'infini, sont euclidiennes ou asymptotiquement hyperboliques avec courbure $- 1$ . Les ondes planes $E (z, ξ)$ sont des fonctions sur $M$ paramétrées par la racine carrée de l'énergie $z$ et la direction $ξ$ de l'onde, interprétée comme un point à l'infini. Si l'ensemble capté $K$ pour le flot géodésique est de mesure de Liouville nulle, nous montrons que, quand $z \to + \infty$ , $E (z, ξ)$ converge microlocalement vers une certaine mesure $μ_{ξ}$ , en moyenne en $ξ$ et en énergie $z$ sur des intervalles de taille fixe. On exprime la vitesse de convergence vers la limite en fonction de la vitesse de fuite du flot géodésique et de son taux maximal d'expansion. Quand le flot est Axiom A sur $K$ , la vitesse de convergence est une puissance négative de $z$ . Enfin, en guise d'application, nous donnons des développements asymptotiques de type Weyl à plusieurs termes pour les traces locales de projecteurs spectraux, avec un reste dépendant de la vitesse de fuite du flot.

We study microlocal limits of plane waves on noncompact Riemannian manifolds $(M, g)$ which are either Euclidean or asymptotically hyperbolic with curvature $- 1$ near infinity. The plane waves $E (z, ξ)$ are functions on $M$ parametrized by the square root of energy $z$ and the direction of the wave, $ξ$ , interpreted as a point at infinity. If the trapped set $K$ for the geodesic flow has Liouville measure zero, we show that, as $z \to + \infty$ , $E (z, ξ)$ microlocally converges to a measure $μ_{ξ}$ , in average on energy intervals of fixed size, $[z, z + 1]$ , and in $ξ$ . We express the rate of convergence to the limit in terms of the classical escape rate of the geodesic flow and its maximal expansion rate—when the flow is Axiom A on the trapped set, this yields a negative power of $z$ . As an application, we obtain Weyl type asymptotic expansions for local traces of spectral projectors with a remainder controlled in terms of the classical escape rate.

MR Zbl

DOI : 10.24033/asens.2217

Classification : 58J50; 58J40, 30F35, 30F45
Keywords: Semiclassical measures, plane waves, Eisenstein functions, Weyl law.
Mot clés : Mesures semi-classiques, ondes planes, fonctions d'Eisenstein, loi de Weyl.

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     author = {Dyatlov, Semyon and Guillarmou, Colin},
     title = {Microlocal limits of plane waves  and {Eisenstein} functions},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {371--448},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 47},
     number = {2},
     year = {2014},
     doi = {10.24033/asens.2217},
     mrnumber = {3215926},
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Dyatlov, Semyon; Guillarmou, Colin. Microlocal limits of plane waves  and Eisenstein functions. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 2, pp. 371-448. doi : 10.24033/asens.2217. http://www.numdam.org/articles/10.24033/asens.2217/

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