For a closed Riemannian orbifold O, we compare the spectra of the Laplacian, acting on functions or differential forms, to the Neumann spectra of the orbifold with boundary given by a domain U in O whose boundary is a smooth manifold. Generalizing results of several authors, we prove that the metric of O can be perturbed to ensure that the first N eigenvalues of U and O are arbitrarily close to one another. This involves a generalization of the Hodge decomposition to the case of orbifolds with manifold boundary. Using these results, we study the behavior of the Laplace spectrum on functions or forms of a connected sum of two Riemannian orbifolds as one orbifold in the pair is collapsed to a point. We show that the limits of the eigenvalues of the connected sum are equal to those of the noncollapsed orbifold in the pair. In doing so, we prove the existence of a sequence of orbifolds with singular points whose eigenvalue spectra come arbitrarily close to the spectrum of a manifold, and a sequence of manifolds whose eigenvalue spectra come arbitrarily close to the eigenvalue spectrum of an orbifold with singular points. We also consider the question of prescribing the first part of the spectrum of an orientable orbifold.

对于封闭的黎曼球面O，我们将作用于函数或微分形式的Laplacian谱与以O为边界的U域（边界是光滑流形）的球面的Neumann谱进行比较。综合几位作者的结果，我们证明O的度量可以被扰动以确保U和O的前N个特征值彼此任意靠近。这涉及将Hodge分解推广到具有流形边界的球面的情况。利用这些结果，我们研究了拉普拉斯谱对两个黎曼圆球的连接和的函数或形式的行为，因为该对中的一个圆球被折叠到一个点。我们表明，连接和的特征值的限制等于该对中未折叠的双峰的特征值的限制。这样做，我们证明了存在奇异点的球面序列，其特征值谱任意接近流形的谱，并且存在一系列特征值谱的球面序列任意近似于具有奇异点的球体的特征值谱。