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 arisingby endowing $X\cup_{S^{1}}Y$ with a parameter dependent Riemannian metric, suchthat the restriction of the metric to $Y$ converges to the hyperbolic metric asa limit with respect to the parameter. We describe the associated spectral andscattering theory for such a surface. We further show that the case of asurface with hyperbolic cusp can be scatteringly approximated in a certainsense by the above family.
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机译：令$ X $是一个二维光滑流形，边界为$ S ^ {1} $，并且$ Y = [1，\ infty）\ S×{1} $。我们考虑了通过给$ X \ cup_ {S ^ {1}} Y $赋予参数依赖的黎曼度量而产生的一族完整曲面，因此度量对$ Y $的限制收敛到双曲度量作为极限。参数。我们描述了这种表面的相关光谱和散射理论。我们进一步表明，上述族在某种意义上可以在某种意义上零散地近似具有双曲尖的表面情况。
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