Geodesic
Quantum unique ergodicity for Eisenstein series on PSL 2 (ℤ∖PSL 2 (ℝ)
Annales de l'Institut Fourier, Tome 44 (1994) no. 5, pp. 1477-1504

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In this paper we prove microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on PSL 2 (ℤ)∖PSL 2 (ℝ). This generalizes a recent result of W. Luo and P. Sarnak who proves equidistribution for PSL 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 SL 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.

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 PSL 2 (ℤ)∖PSL 2 (ℝ). La preuve utilise les opérateurs de Hecke, et n’est donc valable que pour les sous-groupes de congruence de SL 2 (ℝ).

@article{AIF_1994__44_5_1477_0,
     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://geodesic.mathdoc.fr/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

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