Modified transmission eigenvalues in inverse scattering theory

We consider the scattering of an acoustic plane wave by an inhomogeneous medium of compact support. Our aim is to introduce a new class of target signatures that can be used to detect changes in the material properties of the scattering object from a knowledge of the far field pattern of the scattered field. To this end we introduce a modified far field operator depending on a parameter eta and show that this operator is injective with dense range provided eta is not an eigenvalue of a new problem called the modified transmission eigenvalue problem. It is explained why this class of target signatures is preferable in some ways to previously studied target signatures that are based on scattering resonances, transmission eigenvalues, or Stekloff eigenvalues.