Quantum unique ergodicity for Eisenstein series on \(PSL_ 2(\mathbb{Z}){\setminus}PSL_ 2(\mathbb{R})\). (English) Zbl 0820.11040

Summary: We prove a microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on \(PSL_ 2({\mathbb{Z}})\backslash PSL_ 2({\mathbb{R}})\). This generalizes a recent result of W. Luo and P. Sarnak proving equidistribution for \(PSL_ 2({\mathbb{Z}})\backslash {\mathbb{H}}\). 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 \(SL_ 2({\mathbb{R}})\). 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.


11F72 Spectral theory; trace formulas (e.g., that of Selberg)
43A85 Harmonic analysis on homogeneous spaces
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
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