Selberg's trace formula for the Hecke operator generated by an involution, and the eigenvalues of the Laplace--Beltrami operator on the fundamental domain of the modular group~$PSL(2,\mathbf Z)$

A. B. Venkov

Izvestiya. Mathematics , Tome 12 (1978) no. 3, pp. 448-462

Résumé

In this paper a derivation is given of a generalized Selberg trace formula corresponding to the odd eigenfunctions of the Laplace–Beltrami operator in the space $L_2(\Gamma\setminus H)$, where the discrete group $\Gamma$ is $\Gamma=PSL(2,\mathbf Z)$ and $H$ is the upper halfplane (the Dirichlet problem on half of the fundamental domain). As an application a generalization is obtained of Minakshisundaram's formula: \begin{equation} \int_0^\infty e^{-t\lambda}\,d\alpha(\lambda)=\frac1t\cdot\frac1{24}+\frac{\ln t}{\sqrt t}\cdot\frac1{8\sqrt\pi}+\frac1{\sqrt t}\cdot\frac1{8\sqrt\pi}(\mathbf C-\ln2)+O_{t\to0,t>0} \end{equation} ($\alpha(\lambda)$ is the corresponding spectral density; $\mathbf C$ is Euler's constant) and also an asymptotic formula characterizing the irregularity of the distribution of the eigenvalues. Similar results are also obtained for all the eigenvalues of the discrete spectrum of the Laplace–Beltrami operator in the space $L_2(\Gamma\setminus H)$ when $\Gamma$ is the indicated group. Bibliography: 18 titles.