Quantum unique ergodicity for Eisenstein series on P S L 2 ( ℤ ∖ P S L 2 ( ℝ )

Dmitry Jakobson

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 5, page 1477-1504
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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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 44.5 (1994): 1477-1504. <http://eudml.org/doc/75106>.

@article{Jakobson1994,
abstract = {In this paper we prove microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on $PSL_2(\{\Bbb Z\})\backslash PSL_2(\{\Bbb R\})$. This generalizes a recent result of W. Luo and P. Sarnak who proves equidistribution for $PSL_2(\{\Bbb Z\})\backslash \{\Bbb H\}$. 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 $SL_2(\{\Bbb R\})$. 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.},
author = {Jakobson, Dmitry},
journal = {Annales de l'institut Fourier},
keywords = {microlocal equidistribution theorem; Wigner distributions; congruence subgroups},
language = {eng},
number = {5},
pages = {1477-1504},
publisher = {Association des Annales de l'Institut Fourier},
title = {Quantum unique ergodicity for Eisenstein series on $PSL_2(\{\mathbb \{Z\}\}\backslash PSL_2(\{\mathbb \{R\}\})$},
url = {http://eudml.org/doc/75106},
volume = {44},
year = {1994},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 5
SP - 1477
EP - 1504
AB - In this paper we prove microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on $PSL_2({\Bbb Z})\backslash PSL_2({\Bbb R})$. This generalizes a recent result of W. Luo and P. Sarnak who proves equidistribution for $PSL_2({\Bbb Z})\backslash {\Bbb H}$. 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 $SL_2({\Bbb R})$. 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.
LA - eng
KW - microlocal equidistribution theorem; Wigner distributions; congruence subgroups
UR - http://eudml.org/doc/75106
ER -

References

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