# zbMATH — the first resource for mathematics

Equidistribution of cusp forms on $$\text{PSL}_ 2({\mathbb{Z}})\setminus \text{PSL}_ 2({\mathbb{R}})$$. (English) Zbl 0868.43011
Summary: We prove a microlocal version of the equidistribution theorem for Wigner distributions associated to cusp forms on $$\text{PSL}_2({\mathbb{Z}})\setminus \text{PSL}_2 ({\mathbb{R}})$$. This generalizes a recent result of W. Luo and P. Sarnak [Publ. Math., Inst. Hautes Étud. Sci. 81, 207-237 (1995; Zbl 0852.11024)] who prove equidistribution on $$\text{PSL}_2({\mathbb{Z}})\setminus {\mathbb{H}}$$.

##### MSC:
 43A85 Harmonic analysis on homogeneous spaces 11F11 Holomorphic modular forms of integral weight 81Q50 Quantum chaos 58C40 Spectral theory; eigenvalue problems on manifolds
Full Text:
##### References:
 [1] Y. COLIN DE VERDIÈRE, Ergodicité et fonctions propres du laplacien, Comm. Math. Phys, 102 (1985), 497-502. · Zbl 0592.58050 [2] I.S. GRADSHTEYN and I.M. RYZHIK, Tables of integrals, Series and Products, Academic Press, 4th edition, 1980. · Zbl 0521.33001 [3] D. JAKOBSON, Quantum unique ergodicity for Eisenstein series on PSL2(ℤ) PSL2(ℝ), Annales de l’Institut Fourier, 44-5 (1994), 1477-1504. · Zbl 0820.11040 [4] D. JAKOBSON, Thesis, Princeton University, 1995. [5] 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. · Zbl 0427.10016 [6] M. LUO and P. SARNAK. Quantum ergodicity of eigenfunctions on PSL2(ℤ)\ℍ, IHES Publ., 81 (1995), 207-237. · Zbl 0852.11024 [7] W. MAGNUS, F. OBERHETTINGER and R.P. SONI, Formulas and theorems for the special functions of mathematical physics, Springer, 1966. · Zbl 0143.08502 [8] A.I. SHNIRELMAN, Ergodic properties of eigenfunctions, Uspekhi Mat. Nauk, 29-6 (1974), 181-182. [9] 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] J. SLATER, Generalized hypergeometric functions, Cambridge Univ. Press, 1966. · Zbl 0135.28101 [11] S. ZELDITCH, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. Journal, 55 (1987), 919-941. · Zbl 0643.58029 [12] S. ZELDITCH, Mean Lindelöf hypothesis and equidistribution of cusp forms and Eisenstein series, Jour. of Funct. Analysis, 97 (1991), 1-49. · Zbl 0743.58034 [13] S. ZELDITCH, Selberg trace formulas and equidistribution theorems for closed geodesics and Laplace eigenfunctions: finite area surfaces, Mem. AMS, 90 (No. 465), 1992. · Zbl 0753.11023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.