  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.
