Annales Henri Poincaré

, Volume 19, Issue 5, pp 1489–1505 | Cite as

A Scattering Approach to a Surface with Hyperbolic Cusp

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Abstract

Let X be a two-dimensional smooth manifold with boundary \(S^{1}\) and \(Y=[1,\infty )\times S^{1}\). We consider a family of complete surfaces arising by endowing \(X\cup _{S^{1}}Y\) with a parameter-dependent Riemannian metric, such that the restriction of the metric to Y converges to the hyperbolic metric as a limit with respect to the parameter. We describe the associated spectral and scattering theory of the Laplacian for such a surface. We further show that on Y the zero \(S^{1}\)-Fourier coefficient of the generalized eigenfunction of this Laplacian, as a family with respect to the parameter, approximates in a certain sense, for large values of the spectral parameter, the zero \(S^{1}\)-Fourier coefficient of the generalized eigenfunction of the Laplacian for the case of a surface with hyperbolic cusp.

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Authors and Affiliations

  1. 1.Institut für AnalysisLeibniz Universität HannoverHannoverGermany

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