T-matrix scaling relations: Necessary and sufficient conditions for the existence of dynamically invariant scaling coefficients

It is well established that a multiplicative in internal coordinates transition operator is a sufficient condition for the existence of dynamically invariant scaling coefficients. Here we show that by defining the dynamical invariance of the scaling coefficients with respect to a Lippman–Schwinger equation of motion, the same multiplicative in internal coordinates transition operator arises as a necessary condition.