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Zbl 0852.11024
Luo, Wenzhi; Sarnak, Peter
Quantum ergodicity of eigenfunctions on $\text {PSL}\sb 2(\bbfZ) \backslash H\sp 2$. (English)
[J] Publ. Math., Inst. Hautes Étud. Sci. 81, 207-237 (1995). ISSN 0073-8301; ISSN 1618-1913/e

The quantum ergodicity in the title concerns distribution results for measures $|f(z) |^2 d\mu(z)$ on the upper half plane $H^2$ modulo the full modular group $\Gamma= \text {PSL}_2 (\bbfZ)$, where $f$ is either a Maass cusp form, or an Eisenstein series with its spectral parameter on the critical line. $d\mu (z)$ denotes the invariant measure $y^{-2} dx dy$. \par Let $u_1, u_2, \dots$ be a maximal orthogonal system of Maass cusp forms, with corresponding eigenvalues $\lambda_1, \lambda_2, \dots\ $. It is shown that for smooth integrable functions $F$ on $X= \Gamma \setminus H^2$: $$\sum_{\lambda_j\leq x} \ \Biggl|\int_X F|u_j |^2 d\mu- {\textstyle {3\over \pi}} \int_X F d\mu \Biggr|^2 \ll_\varepsilon C_F x^{1/2+ \varepsilon} \qquad (x\to \infty)$$ for each $\varepsilon> 0$. The constant $C_F$ depends on supremum norms of partial derivatives of $F$ up to order 8. \par Let $E(z, s)$ be the Eisenstein series of weight zero, with eigenvalue $s- s^2$. As $X$ has infinite mass for the measure $|E(z, 1/2+ it)|^2 d\mu (z)$, the quantum ergodicity is formulated in terms of compact Jordan measurable subsets of $X$. It is shown that for each such subset $A$: $$\int_A |E(z, 1/2+ it)|^2 d\mu (z)\sim {\textstyle {48 \over \pi}} \mu(A) \log t \qquad (t\to \infty).$$ Other results in the paper are an improvement in the error term of the prime geodesic theorem for $X$, and the estimate $$\sum_{\lambda_j\leq x} \sup_B \Biggl|\int_B |u_j (z) |^2 d\mu (z)- {\textstyle {3\over \pi}} \mu (B) \Biggr|^2 \ll x^{20/ 21+ \varepsilon} \qquad (x\to \infty).$$ $B \subset X$ runs over the injective geodesic circles in $X$. \par The proofs use many techniques and results of the theory of real analytic modular forms: $L$-functions associated to modular forms, approximation by Poincaré series, Hecke operators, the Kuznetsov sum formula, and bounds on Fourier coefficients of Maass cusp forms.
[R.W.Bruggeman (Utrecht)]

MSC 2000:: *11F37 Forms of half-integer weight, etc.
11F72 Spectral theory
11F66 Dirichlet series and functional equations related to modular forms
11F30 Fourier coefficients of automorphic forms

Keywords: quantum ergodicity of eigenfunctions; Maass cusp form; Eisenstein series; prime geodesic theorem; modular forms; $L$-functions; Fourier coefficients

Cited in: Zbl 1253.11062 Zbl 0988.11020 Zbl 0982.11030 Zbl 0917.11019 Zbl 0868.43011

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