Mean Lindelöf hypothesis and equidistribution of cusp forms and Eisenstein series (Q1174996)

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.