We construct a surface with a cylindrical end which has a finite number of Laplace eigenvalues embedded in its continuous spectrum. The surface is obtained by attaching a cylindrical end to a hyperbolic torus with a hole. To our knowledge, this is the first example of a manifold with a cylindrical end whose number of eigenvalues is known to be finite and nonzero. The construction can be varied to give examples with arbitrary genus and with an arbitrarily large finite number of eigenvalues. The constructed surfaces also have resonance-free regions near the continuous spectrum and long-time asymptotic expansions of solutions to the wave equation.

中文翻译：

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具有有限和正数嵌入特征值的圆柱端流形

我们构造了一个带有圆柱端的曲面，该曲面在其连续谱中嵌入了有限数量的拉普拉斯特征值。表面是通过将圆柱端连接到带孔的双曲环面来获得的。据我们所知，这是具有圆柱末端的流形的第一个示例，其特征值的数量已知为有限且非零。可以改变构造以给出具有任意属和任意大的有限数量的特征值的示例。构造的表面在连续谱附近也有无共振区域和波动方程解的长时间渐近展开。