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Fuchsian groups and small eigenvalues of the Laplace operator. (Russian. English summary) Zbl 0536.10017
The author considers small eigenvalues \(\lambda_ j\) of the automorphic Laplacian A for a Fuchsian group of the first kind with noncompact fundamental domain. These eigenvalues are outside the continuous spectrum of A. In the paper an estimate of the number of \(\lambda_ j\) is given. Earlier similar results were known for compact Riemannian surfaces (or spaces) only [see P. Buser, Comment. Math. Helv. 52, 25-34 (1977; Zbl 0348.53027); S.-Y. Cheng, ibid. 51, 43-55 (1976; Zbl 0334.35022)].
Reviewer: A.Venkov

11F03 Modular and automorphic functions
11F70 Representation-theoretic methods; automorphic representations over local and global fields
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
35P99 Spectral theory and eigenvalue problems for partial differential equations
37A30 Ergodic theorems, spectral theory, Markov operators
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