Effective adiabatic approximation in the problem of three bodies coupled via short-range potentials

Abstract

The effective adiabatic approximation is constructed for the problem of three bodies on a straight line that are coupled via short-range attractive delta-function potentials. It is shown that, in this system, there arise a nonlocal momentum-dependent long-range effective potential and a polarization potential. A lower bound on the binding energy of the system is obtained to a relative precision of about 10−6. It is shown that, to within 0.03%, this approximation yields a correct asymptotic behavior of solutions and a correct behavior of the phase shift for elastic scattering at relative momenta below the three-body-breakup threshold. A local convergence of the adiabatic expansion in a finite interval of the radial variable is demonstrated.