Inside-outside duality for a class of modified transmission eigenvalues

We introduce a new spectrum associated with the scattering from penetrable objects at a fixed frequency by employing the modified background technique. This spectrum can be used as qualitative indicator on the ma-terial properties of the medium. We prove that, as opposed to the case of classical transmission eigenvalues, the inside-outside duality method allows to reconstruct this spectrum from full aperture far field measurements at a fixed frequency. We derive a necessary and sufficient condition that characterizes the modified transmission eigenvalues using the phase of a modified far field operator constructed form measured data. The main ingredients in the analy-sis are the factorization of the modified far field operator and the asymptotic expansion of the solution in the neighborhood of the eigenvalues. We also prove that one can obtain an approximation of the associated incident waves. We then propose a numerical implementation of the method and validated its efficiency on some synthetic two dimensional examples.