The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a Caputo fractional Laplacian and a variable coefficient wave number $\mu$, as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix-sequences. We report numerical evidences supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed.

翻译：本研究分析了带Caputo分数阶拉普拉斯算子和变系数波数μ的非局部亥姆霍兹方程有限差分近似的渐近谱性质，该方程源于以非局部相互作用和空间变化波速为特征的复杂介质中的波传播问题。具体而言，借助Toeplitz和广义局部Toeplitz理论工具，本文深入研究了非预处理和预处理矩阵序列的谱分析。我们提供的数值实验验证了理论发现。最后，介绍了若干开放性问题及多方向潜在扩展，并进行了简要讨论。