A scattering approach to a surface with hyperbolic cusp

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.

arXiv comment12 pages