Some Results On The Transmission Eigenvalue Problem In Inverse Scattering Theory

The transmission eigenvalue problem dates from the nineteen-eighties, and now it has become an important part of research in inverse scattering theory. In inverse scattering theory, people are concerned about how to determine the properties of the scattering me-dia, and then quantify the presence of abnormalities inside homogeneous media and text the integrity of materials. Therefore, in practical issues, we need to avoid waves without scattering. Since the frequency of the incident wave without scattering corresponds to an eigenvalue of the transmission eigenvalue problem, the transmission eigenvalue problem has attracted wide spread interest from lots of scholar. However, transmission eigenval-ue problem is a non-selfadjoint and nonlinear eigenvalue problem that is not covered by the standard theory of eigenvalue problems for elliptic operators. Therefore, transmission eigenvalue problem is a very important and difficult question in inverse scattering theory.In this present paper, transmission eigenvalue problems in one dimension and trans-mission eigenvalue problems for spherically stratified media are studied. For the one dimension case, we consider the existence of the non-real transmission eigenvalues, the distribution of transmission eigenvalue in the complex plane and the continuous depen-dence of the transmission eigenvalues. In addition, we study the uniqueness problems in the inverse problems, the major idea lies in selecting the least data to ensure the system uniquely.In the first chapter, we introduce the physics backgrounds, the status of the trans-mission eigenvalue problem, and the major work of this paper.In the second chapter, the transmission eigenvalue problem defined on the interval [0,1] is considered. By using Rouche theorem, and the relationships among the zeros of a entire function and its derived function, we give a sufficient conditions of the existence of non-real transmission eigenvalues when n is a function. Furthermore, a sufficient and necessary conditions under which all the transmission eigenvalues are real is given in the case when n is constant.In the third chapter, we concern with the transmission eigenvalue problem where both equations have unknown functions. With the aid of Rouche theorem and the limit property, we discuss the distribution of the transmission eigenvalues on the complex plane and the new existence conditions of the transmission eigenvalues.In the forth chapter, by the theorem on continuity of the zeros of an analytic func- tion, we show that the isolated transmission eigenvalues are continuous functions of the coefficients of the problem and the transmission eigenfunctions depend continuously on the coefficients.In the fifth chapter, we investigate the transmission eigenvalue problem for spher-ically media, and one of the equations just like u"+[λp(t)-q(t)]u=0, that is, two functions p and q in one equation. In the case when ∫01 (?)dt<1, we prove that p and q can be determined uniquely by two sets of transmission eigenvalues. In the case when p satisfies some conditions at the point t=1, we also prove that both p and q can be determined uniquely by two sets of transmission eigenvalues if ∫01 (?)dt=1, and that both p and q can be determined uniquely by two sets of transmission eigenvalues and two sets of normalizing constants ∫1 (?)dt>1.