Abstract

In this paper, we present a spectral-Galerkin method to approximate the zero-index transmission eigenvalues with a conductive boundary condition. This is a new eigenvalue problem derived from the scalar inverse scattering problem for an isotropic media with a conductive boundary condition. In our analysis, we will consider the equivalent fourth-order eigenvalue problem where we establish the convergence when the approximation space is the span of finitely many Dirichlet eigenfunctions for the Laplacian. We establish the convergence rate of the spectral approximation by appealing to Weyl’s law. Numerical examples for computing the eigenvalues and eigenfunctions for the unit disk and unit square are presented. Lastly, we provide a method for estimating the refractive index assuming the conductivity parameter is either sufficiently large or small but otherwise unknown.

中文翻译：

---

具有导电边界和参数估计的零折射率传输特征值的逼近

摘要

在本文中，我们提出了一种谱-Galerkin方法来近似具有导电边界条件的零折射率透射特征值。这是从具有导电边界条件的各向同性介质的标量逆散射问题导出的新特征值问题。在我们的分析中，我们将考虑等效的四阶特征值问题，当逼近空间是拉普拉斯算子的有限多个Dirichlet特征函数的跨度时，我们会建立收敛。我们根据韦尔定律建立光谱逼近的收敛速度。给出了计算单位圆盘和单位平方的特征值和特征函数的数值示例。最后，