Eigenfunctions of the Nuclear Pairing Hamiltonian

Abstract

An examination of the approximation of separability is made for the pairing Hamiltonian $H=\Sigma \stackrel{}{k>\mathrm{}0\mathrm{}}{\epsilon }_k({a_k}^{†}{a}_k+{a_{-k}}^{†}{a}_{-k})-G\Sigma \stackrel{}{k>\mathrm{}0,l>\mathrm{}0\mathrm{}}{a_k}^{†}{a_{-k}}^{†}{a}_{-l}{a}_l;$ separability being defined as the decomposition of the amplitude of any configuration in the eigenfunctions of the pairing Hamiltonian as a product of factors, one factor associated with each occupied level. This approximation is found to be inadequate for the single-particle energy-level spacings and values of $G$ typically used in nuclear calculations. A somewhat less restrictive approximation is introduced which leads to considerably improved solutions of the pairing Hamiltonian. The results of the approximate calculations are compared with available exact solutions and the agreement is found to be extraordinarily good.