Solvability of eigenvalues in image configurations

Eigenvalues of eigenstates in image configurations (n identical nucleons in the j-orbit) are functions of two-body energies. In some cases they are linear combinations of two-body energies whose coefficients are independent of the interaction and are rational non-negative numbers. It is shown here that a state which is an eigenstate of any two-body interaction has this solvability property. This includes, in particular, any state with spin J if there are no other states with this J in the image configuration. It is also shown that eigenstates with solvable eigenvalues have definite seniority v and thus, exhibit partial dynamical symmetry. Most of the derivations apply also to states of image nucleons with image.