Equidistribution of cusp forms on PSL 2 ( 𝐙 ) ∖ PSL 2 ( 𝐑 )

Dmitri Jakobson

Annales de l'institut Fourier (1997)

  • Volume: 47, Issue: 3, page 967-984
  • ISSN: 0373-0956

Abstract

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We prove a microlocal version of the equidistribution theorem for Wigner distributions associated to cusp forms on PSL 2 ( Z ) ∖ PSL 2 ( R ) . This generalizes a recent result of W. Luo and P. Sarnak who prove equidistribution on PSL 2 ( Z ) ∖ H .

How to cite

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Jakobson, Dmitri. "Equidistribution of cusp forms on ${\rm PSL}_2({\bf Z})\backslash {\rm PSL}_2({\bf R})$." Annales de l'institut Fourier 47.3 (1997): 967-984. <http://eudml.org/doc/75252>.

@article{Jakobson1997,
abstract = {We prove a microlocal version of the equidistribution theorem for Wigner distributions associated to cusp forms on $\{\rm PSL\}_2(\{\bf Z\})\backslash \{\rm PSL\}_2(\{\bf R\})$. This generalizes a recent result of W. Luo and P. Sarnak who prove equidistribution on $\{\rm PSL\}_2(\{\bf Z\})\backslash \{\bf H\}$.},
author = {Jakobson, Dmitri},
journal = {Annales de l'institut Fourier},
keywords = {Eisenstein series; cusp forms; Wigner function; -functions; generalized hypergeometric functions; equidistribution theorem; Wigner distributions},
language = {eng},
number = {3},
pages = {967-984},
publisher = {Association des Annales de l'Institut Fourier},
title = {Equidistribution of cusp forms on $\{\rm PSL\}_2(\{\bf Z\})\backslash \{\rm PSL\}_2(\{\bf R\})$},
url = {http://eudml.org/doc/75252},
volume = {47},
year = {1997},
}

TY - JOUR
AU - Jakobson, Dmitri
TI - Equidistribution of cusp forms on ${\rm PSL}_2({\bf Z})\backslash {\rm PSL}_2({\bf R})$
JO - Annales de l'institut Fourier
PY - 1997
PB - Association des Annales de l'Institut Fourier
VL - 47
IS - 3
SP - 967
EP - 984
AB - We prove a microlocal version of the equidistribution theorem for Wigner distributions associated to cusp forms on ${\rm PSL}_2({\bf Z})\backslash {\rm PSL}_2({\bf R})$. This generalizes a recent result of W. Luo and P. Sarnak who prove equidistribution on ${\rm PSL}_2({\bf Z})\backslash {\bf H}$.
LA - eng
KW - Eisenstein series; cusp forms; Wigner function; -functions; generalized hypergeometric functions; equidistribution theorem; Wigner distributions
UR - http://eudml.org/doc/75252
ER -

References

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  1. [CdV] Y. COLIN DE VERDIÈRE, Ergodicité et fonctions propres du laplacien, Comm. Math. Phys, 102 (1985), 497-502. Zbl0592.58050MR87d:58145
  2. [GR] I.S. GRADSHTEYN and I.M. RYZHIK, Tables of Integrals, Series and Products, Academic Press, 4th edition, 1980. Zbl0521.33001
  3. [Ja94] D. JAKOBSON, Quantum Unique Ergodicity for Eisenstein Series on PSL2(ℤ) PSL2(ℝ), Annales de l'Institut Fourier, 44-5 (1994), 1477-1504. Zbl0820.11040MR96b:11068
  4. [Ja95] D. JAKOBSON, Thesis, Princeton University, 1995. 
  5. [Kuz] N. KUZNETSOV, Peterson's conjecture for cusp forms of weight zero and Linnik's conjecture; sums of Kloosterman sums, Mat. Sb., 111 (1980), 334-383. Zbl0427.10016MR81m:10053
  6. [LS] M. LUO and P. SARNAK. Quantum Ergodicity of Eigenfunctions on PSL2(ℤ), IHES Publ., 81 (1995), 207-237. Zbl0852.11024
  7. [MOS] W. MAGNUS, F. OBERHETTINGER and R.P. SONI, Formulas and theorems for the special functions of mathematical physics, Springer, 1966. Zbl0143.08502MR38 #1291
  8. [Sn74] A.I. SHNIRELMAN, Ergodic Properties of Eigenfunctions, Uspekhi Mat. Nauk, 29-6 (1974), 181-182. 
  9. [Sn93] 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, Springer, 1993. 
  10. [S1] J. SLATER, Generalized hypergeometric functions, Cambridge Univ. Press, 1966. Zbl0135.28101
  11. [Ze87] S. ZELDITCH, Uniform distribution of Eigenfunctions on compact hyperbolic surfaces, Duke Math. Journal, 55 (1987), 919-941. Zbl0643.58029MR89d:58129
  12. [Ze91] S. ZELDITCH, Mean Lindelöf hypothesis and equidistribution of cusp forms and Eisenstein series, Jour. of Funct. Analysis, 97 (1991), 1-49. Zbl0743.58034MR92h:11046
  13. [Ze92] S. ZELDITCH, Selberg Trace Formulas and Equidistribution Theorems for Closed Geodesics and Laplace Eigenfunctions: Finite Area Surfaces, Mem. AMS, 90 (No. 465), 1992. Zbl0753.11023

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