Interaction of $N$ charged particles in the frame of the modified $\mathrm{ВВК}$ approximation: $(N-1)$-particle cluster and a distant particle

In this paper, we consider the problem of $N$ three-dimensional quantum particles with pair potentials,slowly (in the Coulomb manner) decreasing at infinity. We show that the description of the dynamics of an $N-1$-particle localized subsystem and a distant particle interacting with it is constructed on the basis of a certain quantization procedure of coordinates in the subsystem. We assume in this case that the state of the $N-1$-particle localized subsystem itself is completely known, regardless of whether the subsystem is in a bound state (cluster) or not (quasi-cluster). The quality of the asymptotics of the state of the $N$-particle system which we constructed is determined by the descrease rate of the residual of the Schrédinger equation at infinity with respect to the hyperradius of the complete system.