Kohn-Type Variational Principle for Three-Body Breakup Processes

A Kohn-type variational principle normally requires knowledge of the asymptotic form of the time-reversed final-state wave function. For the breakup collision of a bound pair of particles by a third particle, the time-reversed final state is a state in which three free particles are incident, and the asymptotic form of the associated wave function is only incompletely understood. We have nevertheless been able to obtain a Kohn-type variational principle for the breakup scattering amplitude (for short-range potentials). The final expression involves only well-defined integrals, though in the course of the derivation we are forced to separate finite integrals into separately divergent components. These divergences, which also occur elsewhere in scattering theory, are handled by a redefinition of the integrals which is the analog for integrals of Ces\`aro summation.