Cakoni Fioralba, Colton David, Haddar Houssem
Department of Mathematics, Rutgers University, New Brunswick, NJ, USA.
Department of Mathematical Sciences, University of Delaware, Newark, DE, USA.
Proc Math Phys Eng Sci. 2020 Dec;476(2244):20200612. doi: 10.1098/rspa.2020.0612. Epub 2020 Dec 16.
In this paper, we develop a conceptually unified approach for characterizing and determining scattering poles and interior eigenvalues for a given scattering problem. Our approach explores a duality stemming from interchanging the roles of incident and scattered fields in our analysis. Both sets are related to the kernel of the relative scattering operator mapping incident fields to scattered fields, corresponding to the exterior scattering problem for the interior eigenvalues and the interior scattering problem for scattering poles. Our discussion includes the scattering problem for a Dirichlet obstacle where duality is between scattering poles and Dirichlet eigenvalues, and the inhomogeneous scattering problem where the duality is between scattering poles and transmission eigenvalues. Our new characterization of the scattering poles suggests a numerical method for their computation in terms of scattering data for the corresponding interior scattering problem.
在本文中，我们开发了一种概念上统一的方法，用于刻画和确定给定散射问题的散射极点和内部特征值。我们的方法探索了一种对偶性，这种对偶性源于在分析中交换入射场和散射场的角色。这两组都与将入射场映射到散射场的相对散射算子的核相关，分别对应于内部特征值的外部散射问题和散射极点的内部散射问题。我们的讨论包括狄利克雷障碍物的散射问题，其中对偶性存在于散射极点和狄利克雷特征值之间；以及非均匀散射问题，其中对偶性存在于散射极点和传输特征值之间。我们对散射极点的新刻画提出了一种根据相应内部散射问题的散射数据来计算它们的数值方法。
