Translation Representations for Automorphic Solutions of the Wave Equation in Non-euclidean Spaces; the Case of Finite Volume

Let T be a discrete subgroup of automorphisms of H", with fundamental polyhedron of finite volume, finite number of sides, and N cusps. Denote by Ar the Laplace-Beltrami operator acting on functions automorphic with respect to T. We give a new short proof of the fact that Ar has absolutely continuous spectrum of uniform multiplicity N on (-00,«11 l)/2)2), plus a standard discrete spectrum. We show that this property of the spectrum is unchanged under arbitrary perturbation of the metric on a compact set. Our method avoids Eisenstein series entirely and proceeds instead by constructing explicitly a translation representation for the associated wave equation. Introduction. Using the methods developed in Parts I and II of [5], we obtain a short proof for the existence and completeness of incoming and outgoing translation representations for the wave equation acting on automorphic functions with fundamental polyhedron of finite volume and a finite number of sides. As a by-product we show that the associated Laplace-Beltrami operator has a standard discrete spectrum plus an absolutely continuous spectrum of uniform multiplicity TV on (-00, ((« — l)/2)2), where n is the dimension of the hyperbolic space and TV the number of cusps. We also treat automorphic solutions of a perturbed wave equation when the support of the perturbation is compact. A spectral theory for the Laplace-Beltrami operator in this setting was first obtained by A. Selberg in his seminal paper [7] of 1956. Since then several different approaches to this problem have been developed among which we note that of Faddeev [1] and Faddeev and Pavlov [2]. The latter paper made the connection with the Lax-Phillips theory of scattering which we exploited in our 1976 monograph [3]. All of these treatments arrived at the spectral theory via the Eisenstein functions. In the present paper we attack the problem directly through the translation representation, which can be given by an explicit integral formula; this method was first described in [6]. A more complete discussion of this problem can be found in Venkov [8]. Received by the editors July 6, 1984. 1980 Mathematics Subject Classification. Primary 30F40, 35J15, 47A70; Secondary 10D15. 'The work of the first author was supported in part by the Department of Energy under contract DE-AC02-76 ERO 3077 and the second author by the National Science Foundation under Grant MCS-8304317 and the Danish National Science Research Council under Grant 11-3601. ©1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page