Selberg trace formulae, pseudodifferential operators, and geodesic periods of automorphic forms
Let \(\Gamma\) be a cocompact discrete subgroup of PSL(2,\({\mathbb{R}})\) acting on the upper half-plane \({\mathfrak H}\), and let \(X_{\Gamma}\) be the corresponding Riemann surface. Associated to \(X_{\Gamma}\) are two kinds of data: (a) the spectral data of the Laplacian \(\Delta\), the eigenvalues \(\lambda_ k\) and the corresonding normalized eigenfunctions \(u_ k\); (b) the length spectrum \(\{L_{\gamma}\}\) of lengths of closed geodesics, and the geodesic flow \(G^ t.\) The Selberg trace formula provides a famous link between the data (a) and (b). The aim of the paper under review is to prove a general trace formula so as to provide a link between (a) certain so-called ``pseudomeasures'' \(dU_ k\) on \(C(S^*X_{\Gamma})\) which are related with the eigenfunctions \(u_ k\) via pseudodifferential operator theory and (b) certain probability measures \(d\mu_{\gamma}\) on \(C(S^*X_{\Gamma})\), where \(\gamma\) runs through the closed geodesics on \(X_{\Gamma}.\) The author's procedure is to relate integrals of automorphic forms against \(\{dU_ k\}\) and \(\{d\mu_{\gamma}\}\) via trace formulae. The formulae are too complicated to be given here. By way of application the author proves a weak version of Weyl's law: \[ \sum_{\sqrt{\lambda_ k}\leq \lambda}(Op(a)u_ k,u_ k)={\mathcal O}(\lambda /\log \gamma)\quad as\quad \lambda \to \infty \] if \(a\perp 1\) is an automorphic form. Other applications are concerned with the equidistribution theory for eigenfunctions and Bowen's equidistribution theory of closed geodesics. These will be dealt with in forthcoming publications by the author [see e.g. Selberg trace formula, \(\psi\) do's and equidistribution theories for closed geodesics and Laplace eigenfunctions. Pseudo-differential operators, Proc. Conf. Oberwolfach/Ger. 1986, Lect. Notes Math. 1256, 467-479 (1987; Zbl 0634.58034)].
scientific article; zbMATH DE number 4053730
scientific article