Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces; the case of finite volume
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- by Peter D. Lax and Ralph S. Phillips PDF
- Trans. Amer. Math. Soc. 289 (1985), 715-735 Request permission
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.References
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Additional Information
- © Copyright 1985 American Mathematical Society
- 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