The Born transmission eigenvalue problem (Q2832571)

The authors consider an inverse scattering problem for time-harmonic acoustic waves from inhomogeneous media. The solution of this problem can be obtained by the linear sampling method allowing a qualitative characterization of the penetrable obstacle, such as location and shape, from the knowledge of the scattered fields generated by the interaction of the obstacle and known incident fields. The implementation of this method requires that the so-called interior transmission problem has a discrete set of eigenvalues. In particular, the authors study the spectral properties of this problem in the Born regime, which can be seen as a linearization of the scattering problem around the unitary refractive index. This study is provided for the particular case of refractive indices depending only on radial variable and for the general case. In the first case, the Born transmission eigenvalues are expressed in terms of a characteristic equation giving an explicit existence result for real refractive indices. In the general case, the discreteness of the Born transmission eigenvalues is proved under suitable conditions on the real and imaginary parts of the refractive index.