On renormalization in nonrelativistic interaction models with infinities

The Green's function of two models with nonrelativistic separable interaction giving rise to infinities in the perturbation expansion is studied. These infinities do not arise from the E+i0 limit, but come from the slow falloff behavior of the vertices, modeled after the infinities in Feynman graphs of field theory. Both models are analytically solvable. It is found that the Green's function obtained from summing the renormalized perturbation series is identical to the direct solution of the Green's function, which requires only an intermediate regularization. In the first model the interaction is split in a singular part giving raise to infinities and a regular part. It is shown that the Green's function is the same as the Green's function derived from only the regular part. This effect is similar to the effect occurring in phi4 field theory in 3+1 dimensions, where the phi4 interaction vanishes after renormalization and the S matrix is trivial. The second model is constructed such that parts of the singular interaction survive in the Green's function