Redundant states, reduced potentials, and extra nodes in the radial wave function

First-principle scattering calculations which include antisymmetrization of a projectile with respect to identical particles in the target result in a nonsymmetric nonlocal effective potential. Such a potential can lead to redundant states in the scattering wave function. In this case the potential is required to satisfy a consistency condition. We discuss this condition and the manner in which it can be imposed. We also discuss the replacement of this potential by a reduced symmetric nonlocal effective potential which does not produce redundant states. This reduced potential generates a scattering wave function orthogonal to the redundant states. If the original equation has one redundant state, the phase shift at zero energy is $\pi$, resulting in an extra node in the zero-energy wave function. The reduced effective potential must retain this extra node. This characteristic of the reduced effective potential is illustrated with an example. We show that the extra node produced by the potential in the example comes either from a spurious state or a bound state of that potential.