Solutions of the Bethe-Salpeter andNDEquations

Results of calculations for bound and scattering states in off-mass-shell and on-mass-shell formalisms are compared by considering the solutions of the Bethe-Salpeter equation in the ladder approximation, and those of the exact $\frac{N}{D}$ equations. The case of two scalar equal-mass particles interacting via the exchange of a third scalar particle is examined. An approximation to the Bethe-Salpeter equation is also proposed which in the simple case under consideration reduces to an approximate $\frac{N}{D}$ solution. In general, the values of the coupling constants required to produce a given bound state are found to be larger for the approximate solution or exact $\frac{N}{D}$ equations than for the Bethe-Salpeter equation. The discrepancy increases as the mass of the exchanged particle becomes smaller, giving a nearby singularity. In view of the large coupling constants obtained in the $\frac{N}{D}$ equations, some remarks are made regarding the relevance of these results to the dynamics of strongly interacting particles.