New sets of eigenvalues in inverse scattering for inhomogeneous media and their determination from scattering data

Abstract

In this paper we develop a general mathematical framework to determine interior eigenvalues from a knowledge of the modified far field operator associated with an unknown (anisotropic) inhomogeneity. The modified far field operator is obtained by subtracting from the measured far field operator the computed far field operator corresponding to a well-posed scattering problem depending on one (possibly complex) parameter. Injectivity of this modified far field operator is related to an appropriate eigenvalue problem whose eigenvalues can be determined from the scattering data, and thus can be used to obtain information about material properties of the unknown inhomogeneity. We discuss here two examples of such modification leading to a Steklov eigenvalue problem, and a new type of the transmission eigenvalue problem. We present some numerical examples demonstrating the viability of our method for determining the interior eigenvalues form far field data.

Analysis of two transmission eigenvalue problems with a coated boundary condition

“…One can also artificially imbed the scattering object in a background where the refractive index is negative (see for e.g. [4]). The standard transmission eigenvalue problems are non self-adjoint as well as non-linear where as these modified transmission eigenvalue problems are linear.…”

Modified transmission eigenvalues in inverse scattering theory

“…Applying the boundary conditions (7.1d)-(7.1f) (in a similar manner to [7]) now implies that the coefficients satisfy        (rh (1) n (kr)) | r=1 0 −(rj n (k √ γηr)) | r=1 0 −η −1 n(n + 1) 0 h…”

Transmission eigenvalues with artificial background for explicit material index identification

“…In other words, there is no loss of information in complex eigenvalues which can occur for Problem (4). The proof of Theorem 3.1 is similar to the one of [3,Theorem 7] and uses the fact that (F g, g)…”