Separable Approximations to thetMatrix

We extend Reiner's comparison of several separable approximations for the off-energy-shell $t$-matrix elements $t(p, k; s)$ with those for a local, central square-well potential for negative energy $s$. We treat other separable potentials besides the Noyes approximation considered by Reiner, namely, the Weinberg series terminated at one, two, or three terms, and the unitary-pole approximation. We also consider a larger range of momenta and energy than treated by Reiner. We find that the unitary-pole approximation is in general the best of the four one-term separable approximations used. The off-energy-shell values of $t(p, k; s)$ have an error of less than 5% of the value of $t(0, 0; 0)$. The Weinberg series converges rapidly, two terms giving in general an error of less than 1% of $t(0, 0; 0)$. We further compare phase shifts for positive energy and the effective-range parameters. By definition, the Noyes prescription gives the exact value. The unitary-pole approximation gives good results, and again the two-term Weinberg series is very satisfactory. We discuss qualitatively problems arising in the more realistic case (for the two-nucleon potential) from the use of other shapes for the attractive potential, of a strong short-range repulsion, and of tensor forces.