We consider the eigenvalue problem for the restricted fractional Laplacian in a bounded domain with homogeneous Dirichlet boundary conditions. We introduce the notion of fractional capacity for compact subsets, with the property that the eigenvalues are not affected by the removal of zero fractional capacity sets. Given a simple eigenvalue, we remove from the domain a family of compact sets which are concentrating to a set of zero fractional capacity and we detect the asymptotic expansion of the eigenvalue variation; this expansion depends on the eigenfunction associated to the limit eigenvalue. Finally, we study the case in which the family of compact sets is concentrating to a point.

中文翻译：

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关于去除小分数容量集的分数拉普拉斯算子的简单特征值

我们考虑具有齐次狄利克雷边界条件的有界域中受限分数拉普拉斯算子的特征值问题。我们引入了紧凑子集的分数容量的概念，其特性是特征值不受零分数容量集的去除的影响。给定一个简单的特征值，我们从域中删除一组紧集，这些紧集集中到一组零分数容量，我们检测特征值变化的渐近扩展；这种展开取决于与极限特征值相关的特征函数。最后，我们研究紧集族集中到一个点的情况。