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With the use of the stereographic projection of momentum space into thefour-dimensional sphere of unit radius. the possibility of the analyticalsolution of the three-dimensional two-body Lippmann-Schwinger equation with theCoulomb interaction at negative energy has been studied. Simple analyticalexpressions for the three-dimensional Coulomb transition matrix in the case ofthe repulsive Coulomb interaction and positive integer values of the Coulombparameter have been obtained. The worked out method has been also applied forthe generalized three-dimensional Coulomb transition matrix in the case of theattractive Coulomb interaction and negative integer values of the Coulombparameter.

Coulomb transition matrix at negative energy and integer values of interaction parameter