Numerical Study of the 8-32-Particle Eigenstates of the Pairing Hamiltonian

Exact eigenstates of the pairing-force Hamiltonian have been calculated for $2N$ particles in $2N$ equispaced, doubly degenerate single-particle levels for $N=4, 8, 12, \mathrm{and} 16$. The states calculated are the ground state and the first two excited seniority-zero and seniority-two states. These states were calculated for values of the interaction strength that give pairing energies that are typical of nuclear systems. Numerical results are given for the energies and occupation probabilities of these states and are compared with the corresponding results of the Bardeen-Cooper-Schrieffer (BCS) theory. It is shown that the BCS theory is a worse approximation for the larger values of $N$ than it is for the smaller values, with errors as high as 140% (20%) for the excitation energies of the seniority-zero (-two) states. The methods used in the calculation of these states are outlined.