Test of the Boundary-Condition Constraint Method for Nuclear Reactions

The boundary-condition constraint method (BCCM) is a generalization of $R$-matrix theory that permits the shell model to be used as a basis for nuclear-reaction calculations and which permits the correct farasymptotic behavior of the wave function to be imposed as a constraint on shell-model bound-state calculations. For practical application it is necessary to truncate sums over an infinite number of levels. The purpose of this paper is to investigate how effective the few-level approximation might be. The BCCM is used to calculate an $s$-wave bound state for a square well in the two- and three-level approximations. The BCCM results are found to compare favorably with the exact and shell-model results. The BCCM is used to calculate the $s$ and $p$-wave scattering from a square well in the two- and three-level approximations. The results are found to compare favorably with the exact and Wigner $R$-matrix-theory results. We note that within the $n$-level approximation there is some ambiguity in the form of the Green's function to be used with the BCCM.