An invariance theorem in acoustic scattering theory

Karp's theorem states that if the far-field pattern corresponding to the scattering of a time-harmonic acoustic plane wave by a sound-soft obstacle is invariant under the group of orthogonal transformations in R(3) (rotations in R(2)), then the scatterer is a sphere (circle). The theorem is generalized to the case where the invariant group of the far field pattern is only a subgroup of the orthogonal group, and for a class of mixed boundary conditions.