By using the (in general complex) phase shifts for elastic neutron-deuteron scattering in the quartet channel, calculated by means of the exact few-body theory, as input in the fixed-I inverse scattering theory of Marchenko, we construct equivalent local two-body potentials. The latter, though being complex and energy independent, nevertheless reproduce the input phase shifts, which are real below and complex above the deuteron breakup threshold, to a high accuracy. It is even more remarkable that the scattering wave functions (with real boundary conditions) for energies in the elastic regime, calculated with these potentials, are real outside the range of the force, and to a very good approximation real also in the interior region.
The real part of the phase shift for elastic neutron-deuteron scattering in the quartet 5-wave channel, as calculated with the exact three-body theory, assumes at threshold the value x if normalized to zero at infinity; that is, it does not comply with the expectations raised by a naive application of Levinson's theorem since no bound state exists in this channel. A description of this situation on an equivalent two-body level via a potential, constructed by means of the Marchenko inverse scattering theory, necessitates the introduction of a fictitious bound state. This predominantly attractive, equivalent local potential can be related via supersymmetry to a strictly phase equivalent partner potential. The latter is unique and purely repulsive, a behavior already exhibited by the underlying exact eH'ective neutron-deuteron interaction. At the origin it possesses a singularity of the centrifugal barrier type which admits of the required zero-energy phase shift value of x by means of a modified version of Levinson's theorem. Hence, the unphysical bound state of the attractive equivalent local potential plays a role in three-body scattering theory analogous to the one of a Pauli-forbidden state in the context of the resonating group method.
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