Plane-wave approximation in theS-matrix and the construction of bound-state wave functions

Abstract

Three inversion problem approaches — byGelfand-Levitan, Marchenko andPetras [5] —in both non-relativistic and relativistic (Klein-Gordon) variants are used in an approximation scheme selected to construct bound-state wave functions which are advantageous for purposes of model hadron physics. This family of wave functions is created exclusively by theS-matrix quantities and derived in the approximation which requires the Jost function to be equal to the unity throughout the continuous spectrum (the plane-wave approximation). As a consequence of the difference in boundary conditions of the mentioned approaches, the resulting approximate wave functions are not identical, but it is shown that there exists a parallelism as to the form among them. This parallelism is explained more extensively in the non-relativistic case, where the transformation properties of alternative sets of functions are treated. In the present paper it is demonstrated that in the relativistic variants of the above approaches the non-relativistic plane-wave-approximation form of the constructed wave functions for a given bound state is preserved.