Use of the Generalized Jost Function in Quantum Field Theory

The structure of the S-matrix is investigated in terms of the so-called generalized Jost function. Starting from the partial wave dispersion relation, the analytic properties of this function can be derived in the single-channel scattering, and it is shown that there exists a close connection between the analyticity of this function and the generalized Levinson relation. In the case where the one-particle singularity in scattering amplitudes is caused by the elementary particle, the corresponding Jost function has a pole, while in the case where this singularity is caused by the composite bound) state, the Jost function does not have the pole. Singularities corresponding to the Castillejo- Dalitz-Dyson ambiguity are completely separated. Thus, by making the conjecture that the analyticity in the low-momentum region still remains in the multichannel case, the Levinson theorem can be extended to the multichannel case and to the S- matrix theory where the Hamiltonian is not used explicitly. A new representation of the S-matrix, and how the structure of the Smatrix changes according to each case mentioned above, is clarified.