The inverse scattering problem for anisotropic media with data measured inside a cavity is considered. We aim to introduce the exterior Steklov eigenvalues (ESEs) and the modified exterior Steklov eigenvalues (MESEs) to detect changes in the material properties of the medium from a knowledge of a modified near field operator each. First, a connection between a modified near field operator and the exterior Steklov eigenvalue problem is established. Then the ESEs are proved to be discrete and lie in the lower half-plane. Since the existence of the ESEs is unknown due to non-self-adjointness, we propose the modified exterior Steklov eigenvalue problem for which infinitely many eigenvalues exist. Based on the factorizations of both the modified near field operator and the auxiliary near field operator, a nonsymmetric version of generalized linear sampling method (GLSM) is employed to compute the ESEs. This method also allows for the computation of the MESEs by adjusting the definiti...

中文翻译：

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逆散射中的外部Steklov特征值和修正的外部Steklov特征值

考虑了各向异性介质的反散射问题，该各向异性介质具有在腔体内测量的数据。我们旨在介绍外部Steklov特征值（ESE）和修改后的外部Steklov特征值（MESE），以根据各自的修改近场算子的知识来检测介质的材料特性的变化。首先，建立改进的近场算子与外部Steklov特征值问题之间的联系。然后证明ESE是离散的并且位于下半平面。由于非自伴性导致ESE的存在是未知的，因此我们提出了修正的外部Steklov特征值问题，该问题存在无限多个特征值。基于修改后的近场算子和辅助近场算子的因式分解，非对称版本的广义线性采样方法（GLSM）用于计算ESE。该方法还可以通过调整定义来计算MESE。