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Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces; the case of finite volume. (English) Zbl 0576.30037

Let \(\Gamma\) be a cofinite discrete subgroup of automorphisms of n- dimensional hyperbolic Laplacian acting on its natural domain in \(L^ 2(\Gamma \setminus {\mathbb{H}}^ n)\). The work under review contains a short proof for the existence and completeness of the translation representations for the wave equation associated with \(\Gamma\)- automorphic data. In particular, the authors’ analysis yields that \(\Delta\) has a standard discrete spectrum and an absolutely continuous spectrum on \(]-\infty,((n-1)/2)^ 2]\) of multiplicity equal to the number of cusps of \(\Gamma\). An interesting feature of the present approach is that it arrives at these fundamental resuls of Selberg without using the Eisenstein series. The main tools of the authors are the non-Euclidean wave equation and the energy form.
Reviewer: J.Elstrodt

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
35L05 Wave equation
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
11F03 Modular and automorphic functions
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