Uniqueness of refractive indices and transmission coefficients by an inhomogeneous medium in acoustic scattering

We are concerned with the inverse scattering problem of recovering the refractive indices and transmission coefficients by the corresponding acoustic far-field measurement encoded into the scattering amplitude. Our first uniqueness result is to determine a constant refractive index by the fixed incident direction scattering amplitude, the proof of which is mainly based on the discreteness of the corresponding interior transmission eigenvalues. Then motivated by the previous work (), the second uniqueness result is established to recover a piecewise constant refractive index from the far-field pattern at a fixed frequency.