Equivalence of post and prior sum rules for inclusive breakup reactions

A critical examination of sum rules derived previously by Austern and Vincent (post form) and by Udagawa and Tamura (prior form) demonstrates that agreement between the two approaches is obtained if certain approximations implicit in the Udagawa-Tamura prior-form derivation are avoided. We examine the relation of the two approaches to singularities of the post-form distorted wave Born approximation matrix element and to the procedures for reduction of a many-body theory by use of effective operators in a model space. The two-step heuristic model is seen to be invalid for prior-form inelastic breakup; it is necessary to take account of nuclear excitations during projectile breakup. Careful treatment of the non-Hermiticity of kinetic energy operators with respect to continuum wave functions is required.