The modified transmission eigenvalue problem arises in inverse scattering theory for inhomogeneous media, by embedding the relevant medium into an other, artificially introduced inhomogeneous medium. In the present work, we examine the case where the artificial medium is characterised as a metamaterial, i.e. having a negative valued refractive index. Our aim is to construct an appropriate spectral Galerkin method to compute the modified transmission eigenvalues, with potential use to the inverse spectral problem as well. We show that the modified transmission eigenvalue problem corresponds to a compact and self-adjoint operator for which the eigenfuction system is not complete in the solution space. By introducing an auxiliary Dirichlet–Neumann eigenvalue problem, we construct an eigenfuction system which has the desired completeness property. We use this complete system to define the Galerkin scheme and by applying some results for compact and positive operator eigenvalue problems, we prove the convergence of our method. We present some numerical examples which validate the eigenvalues approximation. Finally, we pose the corresponding inverse spectral problem and show that the largest eigenvalue can determine an unknown constant refractive index.

修正的透射特征值问题出现在非均匀介质的逆散射理论中，方法是将相关介质嵌入到另一种人工引入的非均匀介质中。在目前的工作中，我们研究了人造介质被表征为超材料的情况，即具有负值的折射率。我们的目标是构建一个合适的谱 Galerkin 方法来计算修改后的透射特征值，同时也可能用于逆谱问题。我们表明，修正的传输特征值问题对应于一个紧凑的自伴随算子，其特征函数系统在解空间中是不完备的。通过引入辅助 Dirichlet-Neumann 特征值问题，我们构建了一个具有所需完备性的特征函数系统。我们使用这个完整的系统来定义伽辽金方案，并通过将一些结果应用于紧算子和正算子特征值问题，我们证明了我们方法的收敛性。我们提出了一些验证特征值近似的数值例子。最后，我们提出了相应的逆光谱问题，并表明最大特征值可以确定未知的常数折射率。