Weak asymptotics of the wave function for an $N$-particle system and asymptotic filtration

We construct asymptotic representations for large values of the hyperradius for the scattering wave function of an $N$-particles system regarded as a generalized function of angular coordinates. We express the coefficients of the asymptotic representations in terms of the $N$-particle scattering matrix. We discover the phenomenon of asymptotic filtration: only scattering processes in which all particles are free both before and after interaction contribute to the leading terms of such an asymptotic representation. We use the obtained representations to construct the correct asymptotic forms of the partial components of the $N$-particle wave function in the hyperspherical representation.