Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces; the case of finite volume

Authors:
Peter D. Lax and Ralph S. Phillips

Journal:
Trans. Amer. Math. Soc. **289** (1985), 715-735

MSC:
Primary 11F72; Secondary 35P25, 58G25

DOI:
https://doi.org/10.1090/S0002-9947-1985-0784011-9

MathSciNet review:
784011

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Abstract: Let be a discrete subgroup of automorphisms of , with fundamental polyhedron of finite volume, finite number of sides, and cusps. Denote by the Laplace-Beltrami operator acting on functions automorphic with respect to . We give a new short proof of the fact that has absolutely continuous spectrum of uniform multiplicity on , 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.

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DOI:
https://doi.org/10.1090/S0002-9947-1985-0784011-9

Article copyright:
© Copyright 1985
American Mathematical Society