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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 784011
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Abstract: Let $ \Gamma $ be a discrete subgroup of automorphisms of $ {{\mathbf{H}}^n}$, with fundamental polyhedron of finite volume, finite number of sides, and $ N$ cusps. Denote by $ {\Delta _\Gamma }$ the Laplace-Beltrami operator acting on functions automorphic with respect to $ \Gamma $. We give a new short proof of the fact that $ {\Delta _\Gamma }$ has absolutely continuous spectrum of uniform multiplicity $ N$ on $ ( - \infty ,{((n - 1)/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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0784011-9
PII: S 0002-9947(1985)0784011-9
Article copyright: © Copyright 1985 American Mathematical Society