In this work we review some proposals to define the fractional Laplace operator in two or more spatial variables and we provide their approximations using finite differences or the so-called Matrix Transfer Technique. We study the structure of the resulting large matrices from the spectral viewpoint. In particular, by considering the matrix-sequences involved, we analyze the extreme eigenvalues, we give estimates on conditioning, and we study the spectral distribution in the Weyl sense using the tools of the theory of Generalized Locally Toeplitz matrix-sequences. Furthermore, we give a concise description of the spectral properties when non-constant coefficients come into play. Several numerical experiments are reported and critically discussed.

中文翻译：

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二维分数阶Laplace算子，近似矩阵和相关频谱分析

在这项工作中，我们回顾了一些在两个或多个空间变量中定义分数拉普拉斯算子的建议，并使用有限差分或所谓的矩阵转移技术提供了它们的近似值。我们从光谱的角度研究所得大矩阵的结构。特别是，通过考虑涉及的矩阵序列，我们分析了极端特征值，给出了条件估计，并使用广义局部Toeplitz矩阵序列理论的工具研究了Weyl意义上的光谱分布。此外，当非常数系数起作用时，我们对频谱特性进行了简要描述。报道了几个数值实验并进行了严格的讨论。