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Mean Lindelöf hypothesis and equidistribution of cusp forms and Eisenstein series. (English) Zbl 0743.58034

Let \(H\) be the upper half plane, \(\Gamma\subset G=\text{PSL}_ 2(\mathbb R)\) a discrete cocompact subgroup, \(X=\Gamma\backslash H\) and \(\{\phi_ \lambda\}\) the eigenfunctions of the Laplace operator. Then \[ \frac 1{N(\lambda)}\sum_{\sqrt{-\lambda j}\leq \lambda}|(A\phi_ j,\phi_ j)-\bar\sigma_ A|@>>\lambda\to \infty> 0, \] where \(A\) is a 0th order pseudo-differential operator, \(\sigma_ A\) is the principal symbol, \(d\omega\) is the Liouville measure and \(\bar \sigma_ A=(1/\hbox{vol}(S^*X))\int_{S^*X}\sigma_ A d\omega\). The purpose of this paper is to generalize this uniform distribution theorem to finite area, non-compact hyperbolic surfaces \(X_ \Gamma\). Then \(L^ 2(\Gamma\backslash H)\) takes the form \(L^ 2(\Gamma\backslash H)={^ 0L^ 2}(\Gamma\backslash H)\oplus \Theta\), where \(^ 0L^ 2\) is the cuspidal subspace and \(\Theta=L^ 2_{\hbox{eis}}\oplus L^ 2_{\hbox{res}}\oplus\mathbb C\), \(L^ 2_{\hbox{eis}}\) is spanned by the Eisenstein series, \(L^ 2_{\hbox{res}}\) is a finite dimensional space spanned by residues of Eisenstein series at poles in \(]1/2,1[\) and \(\mathbb C\) denotes the constants. The author shows that the generic cusp function and the generic Eisenstein series tend to become uniformly distributed in the unit sphere bundle as the eigenvalues tend to infinity.

MSC:

58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
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