Variational Approach to the On- and Off-ShellTMatrix

A variational procedure for calculating the two-body $T$ matrix $T_l(p,p^{\prime };s)$ is proposed, and studied numerically for the case of the Reid ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ soft-core potential. The method is based on a variational principle of the Schwinger type, in which the trial functions are themselves off-energy-shell $T$ matrices with fixed $s$ and $p$ (or fixed $s$ and $p^{\prime }$), which are expressed as linear combinations of a convenient basis set. The variationally calculated $T$ matrix turns out to have the interesting form $T=V+V{G}^{\prime }V$, where $G^{\prime }$ is a finite-rank approximation to the full Green's function, of rank equal to the number of basis functions. It also turns out that for potentials of finite rank the approximation is exact, provided that the space spanned by the basis functions includes the form factors of the potential. Numerical results are given for the Reid potential at energies from -50 to 300 MeV, and show good convergence for both on- and off-shell $T$ matrix elements. The nonvariational estimates obtained directly from the trial functions also converge quite well, but less rapidly than the variational results.