Connection between Bound States or Resonances of Two-Particle and Three-Particle Systems

The connection between three-particle resonances or bound states and the two-particle resonances or bound states of their component pairs is examined. A simpler and more general derivation is given of the Lovelace equations for isobar or bound-state scattering. As an example, a one-dimensional model of a system of three identical particles, each pair of which interacts only through a single bound or resonant state of energy ${\nu }_0$, is constructed, and the corresponding off-shell scattering integral equation is obtained. By examining the analytic structure as a function of the off-shell momentum at fixed energy, an expression is obtained for the off-shell scattering amplitude involving explicitly known functions and the solution of a much less singular equation. When ${\nu }_0$ is the position of a resonance pole of the two-particle system, the three-particle denominator function has a pole at an energy $4{\nu }_0$ on the unphysical sheet where denominator zeros are associated with resonances, provided that the two-body resonance has a smooth form factor. This suggests the presence of a three-body resonance in this neighborhood if the two-body resonance is narrow. That is not true for bound states. The relation of this to the expected behavior of more realistic problems is discussed.