Non-L2 Solutions of Exactly Soluble Relativistic Wave Equations and the Lee Model

Abstract

In nonrelativistic potential theory there can exist singularities of the S matrix which are associated with wavefunctions belonging to a non-L2 class. In this paper the non-L2 character of these singularities is shown to persist in several relativistic models: in (1) two schemes, including that of the Klein-Gordon equation and the coupling of a classical relativistic field to an external exponential source, in (2) the pair theory, classical and quantized, with separable interactions, and in (3) the Lee model. For (2) and (3), poles of the ``proper'' Jost function f̄_(k, 0), i.e., the Fredholm determinant for the outgoing scattering state, correspond to non-L2 class solutions of the associated dynamical field equation in coordinate space. These non-L2 solutions will be called ``shadow'' fields. They are of special importance and bear the same relation as the ``shadow'' states in potential theory because they also are of dynamical origin and do not appear in the completeness or unitarity relations by virtue of their non-L2 status.