SEPARABLE APPROXIMATIONS TO T MATRIX

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. © 1969 The American Physical Society.