Construction of the T-matrix from its symmetric part

The symmetric and the anti-symmetric parts of the transition matrix, for an uncoupled state, can be related to each other by certain singular integral equations. Thus the scattering matrix can be described and parametrized in terms of the symmetric part of the T-matrix, instead of a potential function. The off-diagonal elements of the symmetric part are arbitrary except for a weak condition imposed on them by the range of the force, while the diagonal elements are directly related to the experimental phase shifts. These phase shifts also determine the bound state energy, but not its form factor. From a given set of phase shifts for the lth partial wave, one can construct a class of T-matrices having the same diagonal elements, but differing in the off-diagonal elements. The difference between any two T-matrices belonging to the same class is a transparent T-matrix, which has no observable effect in two-body scattering. Methods of constructing transparent interactions are discussed, and their effect in the nuclear many-body problem is briefly mentioned.