Trace formula for compact \(\Gamma \setminus PSL_ 2({\mathbb{R}})\) and the equidistribution theory of closed geodesics
Let \(\Gamma\) be a discrete, cocompact subgroup of hyperbolic elements of \(G=PSL_ 2({\mathbb{R}})\), and let \(X_{\Gamma}\) be the associated hyperbolic surface \(\Gamma\) \(\setminus G/K\). A theorem of \textit{R. Bowen} [Am. J. Math. 94, 413-423 (1972; Zbl 0249.53033)] asserts that the periodic geodesics on the surface \(X_{\Gamma}\) become equidistributed relative to Haar measure, as the period tends to \(\infty\). In the paper under review, the author refines this theorem by giving fairly precise asymptotic expansions for \(\sum_{L_{\gamma}\leq T}\int_{{\bar \gamma}}f\) when f is a Casimir eigenfunction. Here, \(L_{\gamma}\) is the length or period of the closed geodesic \({\bar \gamma}\). The method of proof uses trace formulas for pseudodifferential operators for which the classical Selberg trace formula is a special case. It is rather long and involved, as one might expect since a fairly tight estimate is the goal. There is a good deal of machinery and methodology here that might be useful in a number of other applications.
trace formulas for pseudodifferential operators
scientific article; zbMATH DE number 4123821
scientific article