Let [Formula: see text] be a locally compact separable ultrametric space. Given a measure [Formula: see text] on [Formula: see text] and a function [Formula: see text] defined on the set of all non-singleton balls [Formula: see text] of [Formula: see text], we consider the hierarchical Laplacian [Formula: see text]. The operator [Formula: see text] acts in [Formula: see text] is essentially self-adjoint and has a purely point spectrum. Choosing a sequence [Formula: see text] of i.i.d. random variables, we consider the perturbed function [Formula: see text] and the perturbed hierarchical Laplacian [Formula: see text] Under certain conditions, the density of states [Formula: see text] exists and it is a continuous function. We choose a point [Formula: see text] such that [Formula: see text] and build a sequence of point processes defined by the eigenvalues of [Formula: see text] located in the vicinity of [Formula: see text]. We show that this sequence converges in distribution to the homogeneous Poisson point process with intensity [Formula: see text].

中文翻译：

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与分级拉普拉斯算子谱相关的泊松近似

令 [公式：见正文] 为局部紧致可分超度量空间。给定 [Formula: see text] 上的度量 [Formula: see text] 和在 [Formula: see text] 的所有非单子球 [Formula: see text] 的集合上定义的函数 [Formula: see text]，我们考虑分层拉普拉斯算子[公式：见正文]。[Formula: see text] 中的算子 [Formula: see text] 本质上是自伴随的，并且具有纯点谱。选择iid随机变量的序列[公式：见文]，我们考虑扰动函数[公式：见文]和扰动分层拉普拉斯算子[公式：见文]在一定条件下，状态密度[公式：见文]存在并且是一个连续函数。我们选择一个点 [公式：见正文]，使得 [公式：见文本]并构建由位于[公式：见文本]附近的[公式：见文本]的特征值定义的点过程序列。我们证明了这个序列在分布上收敛到具有强度的齐次泊松点过程[公式：见文本]。