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A recently proposed separable expansion of the $t$ matrix (assuming the potential to be given) is rederived from the Schwinger variational principle. Also, analytic expressions are given for the integrals in the separable expansion if the potential is a superposition of Yukawa potentials, and for suitable choices of the expansion functions. Calculations with two commonly used $S$-wave nucleon-nucleon potentials show that the expansion can converge significantly faster than the unitary pole expansion if the freedom in the choice of the expansion functions is exploited.

A recently proposed degenerate-kernel scheme for solving Fredholm integral equations of the second kind is extended to singular equations of the Lippmann-Schwinger type. Numerical calculations of on- and off-shell t-matrix elements are carried out at positive and negative energies, for the 1S0 Reid soft-core nucleon-nucleon potential. Satisfactory convergence is achieved, even with the simplest version of the s

A recently proposed separable expansion for the t -matrix for local potentials is extended to the coupled channel of nucleon-nucleon interactions. The simple method yields a rather convergent separable expansion, with simple form factors, and the requirements of two particle unitarity and time reversal symmetry are maintained for approximations of any rank. The method is illustrated for the simplified Reid soft-core potential in the 3 S 1 - 3 D 1 channel, and the results compared with a recent calculation of Pieper.

A recently proposed degenerate-kernel scheme for solving Fredholm integral equations of the second kind gave very good results for Lippmann-- Schwinger equations. The method is here extended to give a very simple and practical method for the three-body equations with separable two-body interactions. Numerical calculations of the three-body amplitudes are carried out at positive energies for two simple models. In spite of the more complicated singularities in the kernel as well as in the solutions of the equation, satisfactory convergence is achieved.

A general scheme is developed for constructing finite-rank approximations to thet-matrix. The emphasis is on obtaining approximations that are good in a particular context (specifically, in the kernel of few-body problems), rather than on attaining good pointwise accuracy. The scheme in its most general form makes use of two arbitrary sets of expansion functions, and various choices for these functions are considered, and their virtues assessed. Other expansion methods (Weinberg series, unitary pole expansion, and the expansion of Ernst, Shakin and Thaler) can be obtained from the formalism through special choices of the expansion functions, but the methods proposed here, unlike those methods, do not require the solution of an eigenvalue problem. Numerical calculations, using simple choices for the expansion functions, are carried out for the Reid 1S0 soft-core N-N potential. They show that the convergence of the t-matrix expansion can be very good indeed, especially if it is used in the operator context for which it is designed.

A new parametrization of the three-body $D$ function based in part on numerical investigation is presented. It is correct in both weak coupling and the Efimov limit, and agrees remarkably well with "exact" numerical calculation in a boson model. It has the correct singularity position but not the correct strength at the three-body threshold. In the neutron-deuteron quartet system it agrees very well with exact calculation and casts light on the relative model independence of these calculations. We also show that the Efimov states become virtual states with increasing coupling.

A new parameterization for the three body D-function is presented. It is correct in both weak coupling and in the Efimov limit and agrees remarkably well with “exact” numerical calculations for a boson model and for the neutron-deuteron system.

We know that the two-body scattering amplitude at low energy goes to a constant. Only recently, it has been shown by Amado and Rubin that the three-body scattering amplitude at low energy diverges like $a{E}^{\ensuremath{-}1}+b{E}^{\ensuremath{-}\frac{1}{2}}+c\mathrm{ln}E$, where $E$ is the three-body center-of-mass energy. We here calculate the coefficients $a$, $b$, $c$ in terms of the two-body $s$-wave scattering length and some kinematical factors. We extend our analysis for the four-body case and find that at low energy the four-body scattering amplitude also diverges, and we classify the divergent terms and indicate how to calculate their coefficients. Beyond the four-body problem the scattering amplitude also diverges at low energy, and we indicate what is needed to calculate the coefficients of the divergent terms in general.