Λ10Approximation for Self-Bound Many-Fermion Systems

We investigate several approximations for self-bound many-fermion systems, each of which employs the Green's-function formalism together with a suitable factorization to close the otherwise infinite hierarchy of coupled integral equations. The particular approximations, which attempt to include the important two-body correlations for short-range forces, differ in the choice of propagator $\Lambda$ for two-particle intermediate states in the $T$ matrix used to calculate the two-body correlation function. The advantages and shortcomings of the choice ${\Lambda }_{00}\sim {G_1}^0{G_1}^0$ have been noted previously and are not treated here. We show that the choice ${\Lambda }_{10}\sim {G_1}^0{G}_1$, which appears in a "natural" factorization, is formulated in a manner which preserves much of the simplicity of the ${\Lambda }_{00}$ approximation. (Numerical calculations using the ${\Lambda }_{10}$ approximation are presented by Foster in the following paper.) The ground-state energies for three $\Lambda$ approximations (${\Lambda }_{00}$, ${\Lambda }_{10}$, and ${\Lambda }_{11}\sim G_1{G}_1$) are expanded through third order in the interaction strength to facilitate comparison with each other and with the Brueckner-Goldstone series. Numerical comparison of these third-order contributions for a square-well potential is presented. ${\Lambda }_{10}$ is in dramatically closer agreement with the full perturbation theory (for ${k_F}^{0}c\lesssim 1$) than are the other $\Lambda$ approximations or Brueckner theory.