Exact solution of the mean-field plus separable pairing model reexamined

Exact solution of the nuclear mean-field plus separable pairing model is reexamined. New auxiliary constraints for solving the Bethe ansatz equations of the model are proposed. By using these auxiliary constraints, the Bethe ansatz form of eigenvectors of the mean-field plus separable pairing Hamiltonian with nondegenerate single-particle energies and nondegenerate separable pairing strengths purposed previously is verified. Since the solutions of the model with one- and two-orbit cases are known, verification of the solutions for these two special cases is made. To demonstrate structure and features of the solution, the model with three orbits in the $ds$ shell is taken as a nontrivial example, of which two-pair results and the ground state of the three-pair case are provided explicitly. Since the number of equations involved increases with the number of orbits and pairs, to solve these equations for a large number of orbits and pairs seems still difficult.