Anisotropic Transmission Eigenvalue Problems: Existence, Discreteness, and Dependence on Conductivities

We consider an anisotropic transmission eigenvalue problem arising in the study of inverse scattering for media with two distinct conductivity parameters. In the framework of qualitative reconstruction methods, such as the linear sampling method and the direct sampling method, the absence of scattering plays a central role: regions where scattering does not occur correspond to transmission eigenvalues. Understanding these values is therefore essential in order to develop accurate and reliable reconstructions of the scatterer. In this work, we formulate the corresponding anisotropic transmission eigenvalue problem, establish the existence and discreteness of transmission eigenvalues, and investigate their dependence on the physical conductivity parameters of the medium. We provide numerical examples that illustrate the theoretical results and highlight the role of transmission eigenvalues in guiding qualitative reconstruction methods.