Selberg trace formulae, pseudodifferential operators, and geodesic periods of automorphic forms
type: Item
instance of: scholarly article
title: Selberg trace formulae, pseudodifferential operators, and geodesic periods of automorphic forms
zbMATH Open document ID: 0646.10024
DOI: 10.1215/S0012-7094-88-05613-X
author: Steve Zelditch
published in: Duke Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 35S99
Mathematics Subject Classification ID: 58J40
zbMATH DE Number: 4053730
zbMATH Keywords: Laplacian
zbMATH Keywords: eigenvalues
zbMATH Keywords: length spectrum
zbMATH Keywords: lengths of closed geodesics
zbMATH Keywords: geodesic flow
zbMATH Keywords: Selberg trace formula
zbMATH Keywords: pseudodifferential operator
zbMATH Keywords: automorphic form
zbMATH Keywords: equidistribution
zbMATH Keywords: eigenfunctions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4053730