no. 5
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

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},
     year = {1994},
     publisher = {Association des Annales de l'Institut Fourier},
     volume = {44},
     number = {5},
     doi = {10.5802/aif.1442},
     mrnumber = {96b:11068},
     zbl = {0820.11040},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1442/}
}
TY  - JOUR
AU  - Jakobson, Dmitry
TI  - Quantum unique ergodicity for Eisenstein series on $PSL_2({\mathbb {Z}}\backslash PSL_2({\mathbb {R}})$
JO  - Annales de l'Institut Fourier
PY  - 1994
SP  - 1477
EP  - 1504
VL  - 44
IS  - 5
PB  - Association des Annales de l'Institut Fourier
UR  - https://www.numdam.org/articles/10.5802/aif.1442/
DO  - 10.5802/aif.1442
LA  - en
ID  - AIF_1994__44_5_1477_0
ER  - 
%0 Journal Article
%A Jakobson, Dmitry
%T Quantum unique ergodicity for Eisenstein series on $PSL_2({\mathbb {Z}}\backslash PSL_2({\mathbb {R}})$
%J Annales de l'Institut Fourier
%D 1994
%P 1477-1504
%V 44
%N 5
%I Association des Annales de l'Institut Fourier
%U https://www.numdam.org/articles/10.5802/aif.1442/
%R 10.5802/aif.1442
%G en
%F AIF_1994__44_5_1477_0
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

[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. | Zbl | MR

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

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

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

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

[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. | Zbl | MR

[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. | Zbl | MR

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

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

[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

Cité par Sources :