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

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

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

@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://archive.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://archive.numdam.org/articles/10.5802/aif.1442/

[AS] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, AMS 55, 7th ed., 1968.

[Ba] Bailey, Generalized hypergeometric series, Cambridge Univ. Press, 1935. | Zbl

[CdV] Y. Colin De Verdière, Ergodicité et fonctions propres du laplacien, Comm. Math. Phys., 102 (1985), 497-502. | MR | Zbl

[DRS] W. Duke, Z. Rudnick and P. Sarnak, Density of Integer Points on Affine Homogeneous Varieties, Duke Math. Jour., 71 (1) (1993), 143-179. | MR | Zbl

[Fa] John, D. Fay, Fourier coefficients for a resolvent of a Fuchsian Group, J. für die Reine und Angew, Math., 293 (1977), 143-203. | MR | Zbl

[Fo] G. B. Folland, Harmonic Analysis in Phase Space, AMS Studies, Princeton Univ. Press, 1989. | MR | Zbl

[G] Anton Good, The square mean of Dirichlet series associated with cusp forms, Mathematika, 29 (1982), 278-295. | MR | Zbl

[GR] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 4th ed., Academic Press, 1980.

[K] T. Kubota, Elementary Theory of Eisenstein Series, Kodansha, Ltd., Tokyo and John Wiley & Sons, New York, 1973. | MR | Zbl

[L] S. Lang, SL2(R), Addison-Wesley, 1975.

[LS] M. Luo and P. Sarnak, Quantum Ergodicity of Eigenfunctions on PSL2(Z)\H2, to appear.

[Me] Tom Meurman, The order of the Maass L-function on the critical line, Colloquia mathematica societatis Janos Bolyai 51. Number theory, Budapest (Hungary), (1987), 325-354. | Zbl

[Ro] Walter Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, I, Math. Annalen, 167 (1966), 293-337. | Zbl

[Sa] P. Sarnak, Horocycle flow and Eisenstein series, Comm. Pure Appl. Math., 34 (1981), 719-739. | Zbl

[Sn1] A.I. Shnirelman, Ergodic Properties of Eigenfunctions, Uspekhi Mat. Nauk, 29 (6) (1974), 181-182.

[Sn2] A.I. Shnirelman, On the Asymptotic Properties of Eigenfunctions in the Regions of Chaotic Motions (Addendum to V. F. Lazutkin's book), KAM Theory and Semiclassical Approximations to Eigenfunctions, Springer, 1993.

[Ti] E. Titchmarsh, The Theory of of The Riemann Zeta Function, Oxford, 1951. | MR | Zbl

[Z1] S. Zelditch, Uniform distribution of Eigenfunctions on compact hyperbolic surfaces, Duke Math. Jour., 55 (1987), 919-941. | MR | Zbl

[Z2] S. Zelditch, Mean Lindelöf hypothesis and equidistribution of cusp forms and Eisenstein series, Journal of Functional Analysis, 97 (1991), 1-49. | MR | Zbl

[Z3] S. Zelditch, Selberg Trace Formulas and Equidistribution Theorems for Closed Geodesics and Laplace Eigenfunctions: Finite Area Surfaces, Mem. AMS 90 (N° 465) (1992). | Zbl

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