This work deals with the isogeometric Galerkin discretization of the eigenvalue problem related to the Laplace operator subject to homogeneous Dirichlet boundary conditions on bounded intervals. This paper uses GLT theory to study the behavior of the gap of discrete spectra toward the uniform gap condition needed for the uniform boundary observability/controllability problems. The analysis refers to a regular $B$-spline basis and concave or convex reparametrizations. Under suitable assumptions on the reparametrization transformation, we prove that structure emerges within the distribution of the eigenvalues once we reframe the problem into GLT-symbol analysis. We also demonstrate numerically, that the necessary average gap condition proposed in \cite{bianchi2018spectral} is not equivalent to the uniform gap condition. However, by improving the result in \cite{bianchi2021analysis} we construct sufficient criteria that guarantee the uniform gap property.

翻译：本文研究有界区间上满足齐次狄利克雷边界条件的拉普拉斯算子特征值问题的等几何伽辽金离散化。利用GLT理论分析离散谱间隙向统一间隙条件过渡的行为，该条件对于一致边界可观测性/可控性问题至关重要。分析基于规则$B$-样条基函数及凹/凸重参数化方法。在重参数化变换的适当假设下，证明通过将问题重构为GLT-符号分析后，特征值分布中会涌现结构性特征。数值实验同时表明，文献\cite{bianchi2018spectral}提出的必要平均间隙条件与统一间隙条件不等价。然而，通过改进文献\cite{bianchi2021analysis}的结果，我们构建了保证统一间隙性质的充分判据。