We consider arbitrary noncompact hyperbolic Riemann surfaces of finite area. For such surfaces, we obtain identities relating the discrete spectrum of the Laplace operator to the resonance spectrum (formed by the poles of the scattering matrix). These identities depend on the choice of a test function. We indicate a class of admissible test functions and consider two examples corresponding to specific choices of the test function.

中文翻译：

---

非紧双曲Riemann曲面上Laplace算子的离散谱与共振谱之间的关系

我们考虑有限面积的任意非紧致双曲Riemann曲面。对于这样的表面，我们获得了将拉普拉斯算子的离散光谱与共振光谱（由散射矩阵的极点形成）相关的恒等式。这些身份取决于测试功能的选择。我们指出了可接受的测试函数类别，并考虑了两个与测试函数的特定选择相对应的示例。