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Asymptotic behaviour of the spectral function of the Laplace-Beltrami operator for cocompact discrete subgroups of \(\text{SL}_2(\mathbb{R})\). (English. Russian original) Zbl 0916.11032
Sb. Math. 189, No. 7, 977-990 (1997); translation from Mat. Sb. 189, No. 7, 23-36 (1998).
Let \(\Gamma\) be a cocompact discrete subgroup of \(\text{SL}_2(\mathbb{R})\) acting on the upper half-plane \(\mathbb{H}\), and let \(\chi\) be a one-dimensional representation of \(\Gamma\). The operator \(-\Delta\) acting on \(L^2(\Gamma\setminus \mathbb{H},\chi, y^{-2}dx dy)\) has an orthonormal basis \((\varphi_j)_{j\geq 1}\) of eigenfunctions with eigenvalues \(0\leq \lambda_1\leq\lambda_2\leq\cdots,\lambda_j\to \infty\). The asymptotic distribution of the eigenvalues (including a good error term) is well-known (Weyl’s law). The asymptotic behaviour of the spectral function \[ \theta(z,z'; \lambda)= \sum_{\lambda_j\leq\lambda} \varphi_j(z)\overline{\varphi_j(z')} \] apparently has not been written out so far in the case under consideration. The present paper fills this gap. The proof is based on Selberg’s spectral decomposition of automorphic kernel functions.
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
11F37 Forms of half-integer weight; nonholomorphic modular forms
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