The reaction A $\leftrightarrow$ A + $\mu$ is considered; A is a fixed point scatterer at the origin, and the $\mu$ are relativistic scalar particles (`mesons') interacting with A only at the origin. Generally, both A and $\mu$ are endowed with `extra' structure, for example, isotopic spin. It is shown that the conventional treatment fails unless the scattering is trivial. An alternative approach is proposed; it is shown that the scattering is almost completely determined by the necessary vanishing of certain scalar products (the orthogonality condition). On the basis of this success, the procedure is reinterpreted in terms of the search for a description of the local interaction as a boundary condition, and it is suggested that this is the most natural way of introducing a local interaction with non-trivial scattering. An important simplification is gained by the use of a relativistic meson field with an indefinite metric; it is this which permits an exact treatment of the problem. Only models with S-wave scattering are considered in this paper.