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Zbl 0820.11040
Jakobson, Dmitry
Quantum unique ergodicity for Eisenstein series on $PSL\sb 2(\bbfZ){\setminus}PSL\sb 2(\bbfR)$. (English)
[J] Ann. Inst. Fourier 44, No.5, 1477-1504 (1994). ISSN 0373-0956; ISSN 1777-5310/e

Summary: We prove a microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on $PSL\sb 2({\bbfZ})\backslash PSL\sb 2({\bbfR})$. This generalizes a recent result of W. Luo and P. Sarnak proving equidistribution for $PSL\sb 2({\bbfZ})\backslash {\bbfH}$. 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 $SL\sb 2({\bbfR})$. 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.

MSC 2000:: *11F72 Spectral theory
43A85 Analysis on homogeneous spaces
11F66 Dirichlet series and functional equations related to modular forms

Keywords: microlocal equidistribution theorem; Wigner distributions; congruence subgroups

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