-coherent state, respectively. On the basis of the above conclusion, the intermediate region for the interaction strength is described in terms of the idea of the interpolation and it is shown that the agreement of the result with the exact one is quite good.

DOI: 10.1143/ptp.117.853

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su(2)\otimes su(1,1)

- and the

su(2)\otimes su(2)

F

the strength parameter of the interaction). The factor $\ensuremath{\alpha}$ is found to be small and constant over a wide range of

F

and in this range the deviation of

W

from the Bogoliubov value is small. There is, however, a transition point

{F}_{T}

, beyond which $\ensuremath{\alpha}$ is constant and large and the usual Bogoliubov approximation is invalid for interactions more attractive than that characterized by

{F}_{T}

. A Bogoliubov-like approximation is shown to be quite accurate in this region. It is then shown that $F_{T}^{\ensuremath{-}1}$ is a linear function of the number of particles in the system.

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.