Role of Levinson's theorem in neutron-deuteron quartet S-wave scattering

The real part of the phase shift for elastic neutron-deuteron scattering in the quartet S wave channel, as calculated with the exact three-body theory, assumes at threshold the value π 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 effective 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 π 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