Determining Transmission Eigenvalues of Anisotropic Inhomogeneous Media from Far Field Data

We characterize interior transmission eigenvalues of penetrable anisotropic acoustic scattering objects by a technique known as inside-outside duality. This method has recently been identified to be able to link interior eigenvalues of the penetrable scatterer with the behavior of the eigenvalues of the far field operator for the corresponding exterior time-harmonic scattering problem. A basic ingredient for the resulting connection is a suitable self-adjoint factorization of the far field operator based on wave number-dependent function spaces. Under certain conditions on the anisotropic material coefficients of the scatterer, the inside-outside duality allows to rigorously characterize interior transmission eigenvalues from multi-frequency far field data. This theoretical characterization moreover allows to derive a simple numerical algorithm for the approximation of interior transmission eigenvalues. Since it is merely based on far field data, the resulting eigenvalue solver does not require knowledge on the scatterer or its material coefficient; several numerical examples show its feasibility and accuracy for noisy data.