This work addresses the Galerkin isogeometric discretization of the one-dimensional Laplace eigenvalue problem subject to homogeneous Dirichlet boundary conditions on a bounded interval. We employ GLT theory to analyze the behavior of the eigenfrequencies when a reparametrization is applied to the computational domain. Under suitable assumptions on the reparametrization transformation, we prove that a structured pattern emerges in the distribution of eigenfrequencies when the problem is reframed through GLT-symbol analysis. Additionally, we establish results that refine and extend those of [3], including a uniform discrete Weyl's law. Furthermore, we derive several eigenfrequency estimates by establishing that the symbol exhibits asymptotically linear behavior near zero.

翻译：本研究探讨了有界区间上满足齐次狄利克雷边界条件的一维拉普拉斯特征值问题的伽辽金等几何离散化方法。我们运用GLT理论分析计算域重参数化时特征频率的行为规律。在重参数化变换满足适当假设的前提下，通过GLT符号分析重构该问题时，我们证明特征频率分布会呈现结构化模式。此外，我们完善并拓展了文献[3]的结论，包括建立一致离散外尔定律。通过证明符号在零点附近呈现渐近线性特性，我们进一步推导出若干特征频率估计。