Boundary-Condition-ModelTMatrix

A number of results on the boundary-condition-model (BCM) $T$ matrix are developed. It is shown that the half-off shell $T$ matrix is unique, but the fully off-shell $T$ matrix is not. Further, it is shown that the ambiguity in the $T$ matrix resides in that part of the complete $T$ matrix which is identifiable as the $T$ matrix for the pure BCM, where by pure BCM is meant no forces outside the boundary-condition radius. Three different formulas for the pure BCM $T$ matrix are presented. The first is derived by using the relations that exist between the half-off-shell $T$ matrix and the fully off-shell $T$ matrix for well-behaved potentials, and is found to be separable. The second is taken from the work of Kim and Tubis. The third is derived from a pseudopotential constructed by Hoenig and Lomon. All three agree exactly half off shell, and satisfy the off-shell unitarity relation. Numerical comparisons are given which show that significant differences can occur in the fully off-shell $T$ matrices. An integral- as well as a differential-equation approach are given for finding the contribution to the BCM $T$ matrix from the forces outside the boundary-condition radius. Separable representations for the BCM $T$ matrix are developed, and their usefulness in carrying out calculations on the three-nucleon system is discussed.