Quantum unique ergodicity for Eisenstein series on $PSL_2({\mathbb {Z}}\backslash PSL_2({\mathbb {R}})$

Jakobson, Dmitry

doi:10.5802/aif.1442

Quantum unique ergodicity for Eisenstein series on

P S L_{2} (ℤ ∖ P S L_{2} (ℝ)

Jakobson, Dmitry

Annales de l'Institut Fourier, Tome 44 (1994) no. 5, pp. 1477-1504.

Résumé
Abstract

Nous donnons la preuve d’une version microlocale d’un résultat de W. Luo et P. Sarnak concernant la répartition asymptotique des fonctions de Wigner associées aux séries d’Eisenstein sur $P S L_{2} (ℤ) ∖ P S L_{2} (ℝ)$ . La preuve utilise les opérateurs de Hecke, et n’est donc valable que pour les sous-groupes de congruence de $S L_{2} (ℝ)$ .

In this paper we prove microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on $P S L_{2} (ℤ) ∖ P S L_{2} (ℝ)$ . This generalizes a recent result of W. Luo and P. Sarnak who proves equidistribution for $P S L_{2} (ℤ) ∖ ℍ$ . The averaged versions of these results have been proven by Zelditch for an arbitrary finite-volume surface, but our proof depends essentially on the presence of Hecke operators and works only for congruence subgroups of $S L_{2} (ℝ)$ . In the proof the key estimates come from applying Meurman’s and Good’s results on $L$ -functions associated to holomorphic and Maass cusp forms. One also has to use classical transformation formulas for generalized hypergeometric functions of a unit argument.

MR Zbl | 2 citations dans Numdam

DOI : 10.5802/aif.1442

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     author = {Jakobson, Dmitry},
     title = {Quantum unique ergodicity for {Eisenstein} series on $PSL_2({\mathbb {Z}}\backslash PSL_2({\mathbb {R}})$},
     journal = {Annales de l'Institut Fourier},
     pages = {1477--1504},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {44},
     number = {5},
     year = {1994},
     doi = {10.5802/aif.1442},
     mrnumber = {96b:11068},
     zbl = {0820.11040},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1442/}
}

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Jakobson, Dmitry. Quantum unique ergodicity for Eisenstein series on $PSL_2({\mathbb {Z}}\backslash PSL_2({\mathbb {R}})$. Annales de l'Institut Fourier, Tome 44 (1994) no. 5, pp. 1477-1504. doi : 10.5802/aif.1442. http://www.numdam.org/articles/10.5802/aif.1442/

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