Inverse scattering theory and transmission eigenvalues (Q2833164)

This book introduces the reader to inverse scattering theory and transmission eigenvalues. It is subdivided into five chapters.NEWLINENEWLINEChapter 1 presents the basic ideas of scattering theory for an inhomogeneous media of compact support. Specifically, the associated inverse scattering problem is provided to the reader. Additionally, the far-field operator is introduced as well as the basic theory of ill-posed problems. Furthermore, uniqueness results for inverse scattering problems for both isotropic and anisotropic media are established. This first chapter is the fundament for the next chapters, in which a qualitative approach to inverse scattering theory is developed.NEWLINENEWLINEChapter 2 focuses on the determination of the support of an inhomogeneous media. First, a class of inversion methods (referred to as qualitative methods) are introduced and analyzed. The task is to find the obstacle from measured far-field data without reconstructing physical parameters such as the index of refraction. The algorithms at hand do not need a forward solver and are easy to be implemented. The first section considers the linear sampling method (LSM). A complete analysis is provided for the isotropic case. However, the general case why it works numerically is not fully justified. The next section illustrates in detail the generalized linear sampling method (GLSM) which circumvents the weak points of LSM. The theory for GLSM is provided for full aperture data. The next two sections are dealing with the inf-criterion and the Factorization method with application to the isotropic inverse problem. Section 5 shows a connection between all previously discussed methods and section 6 shows the applicability of all methods for the anisotropic case.NEWLINENEWLINEChapter 3 deals with the interior transmission problem. Typical questions are (a) the Fredholm property and solvability, (b) the discreteness of transmission eigenvalues, (c) the existence of transmission eigenvalues, and (d) the determination of transmission eigenvalues from scattering data. The first section illustrates the solvability and discreteness for the isotropic case and the second section the solvability and discreteness for the anisotropic case.NEWLINENEWLINEChapter 4 is devoted to the existence of transmission eigenvalues for both the isotropic and anisotropic case. Therefore, some analytical tools are given in Section 1. In Section 2, the existence is shown for the isotropic case. Two additional subsections also consider media with voids and some inequalities for transmission eigenvalues. Some remarks for absorbing media are given as well. Section 3 considers the existence of transmission eigenvalues for anisotropic media distinguishing the case \(n=1\) and \(n\neq 1\). Again, inequalities are given. Additionally, it is explained how one can determine transmission eigenvalues from far-field data using (a) LSM, (b) GLSM, and (c) eigenvalues of the far-field operator. Numerical examples are given as well.NEWLINENEWLINEThe last chapter concentrates on inverse spectral problems for transmission eigenvalues. Section 1 considers the spherically stratified media with spherically symmetric eigenfunctions. Section 2 illustrates the more general case where it is assumed that the transmission eigenfunctions are not spherically symmetric. A list of references and an index ends the book.