The Interior Transmission Problem and Inverse Scattering from Inhomogeneous Media

Abstract

This paper is concerned with the class of far field patterns corresponding to the scattering of time harmonic acoustic plane waves by an inhomogeneous medium in a bounded domain B, with refractive index $n(x)$ It has previously been shown that the class of far field patterns is complete in $L_2 (S^2 )$ except at wavenumbers k, which are so-called transmission eigenvalues of the homogeneous interior transmission problem. In this paper the interior transmission problem is studied and, under milder conditions on n than previously used, the set of transmission eigenvalues is shown to be discrete. Also, at points other than transmission eigenvalues, it is shown that the inhomogeneous interior transmission problem is uniquely solvable. This result is of importance in certain methods for solving the inverse scattering problem of determining the function n from the scattered far fields.