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：本研究探讨了具有Caputo分数阶拉普拉斯算子和变系数波数$\mu$的非局部亥姆霍兹方程有限差分离散的渐近谱特性，此类方程出现在考虑波在复杂介质中传播时，其特点是非局部相互作用和空间变化的波速。具体而言，本研究借助Toeplitz理论和广义局部Toeplitz理论工具，深入分析了未经预条件处理和经预条件处理的矩阵序列的谱特性。我们提供了支持理论发现的数值证据。最后，提出并简要讨论了多个方向的开放性问题及潜在扩展方向。