Inverse Spectral Problems for Transmission Eigenvalues
We consider the transmission eigenvalue problem for spherically stratified media. We will show that in this case there exist in general an infinite number of complex transmission eigenvalues and that these eigenvalues lie in a strip parallel to the real axis. We will also show that a knowledge of all the eigenvalues (both real and complex) uniquely determine the spherically stratified index of refraction. We will also obtain similar results for a new class of transmission eigenvalue problems related to scattering problems for sources and receivers located in a cavity bounded by a spherically stratified medium.As with the classical inverse Sturm-Liouville problem,the analysis used in our investigations relies heavily on the theory of entire functions of a complex variable.