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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Communicated by Jan Dereziński.
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Roidos, N. A Scattering Approach to a Surface with Hyperbolic Cusp. Ann. Henri Poincaré 19, 1489–1505 (2018). https://doi.org/10.1007/s00023-018-0669-3
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DOI: https://doi.org/10.1007/s00023-018-0669-3