Weak asymptotics of wave function for N-particle system and asymptotic filtering

Asymptotic representations for large values of the hyperradius are constructed for the scattering wave function of a system of $N$ particles considered as a generalized function of angular variable coordinates. The coefficients of the asymptotic representations are expressed in terms of the $N$-particle scattering matrix. The phenomenon of asymptotic filtration is discovered, which consists in the fact that only scattering processes contribute to the leading terms of such an asymptotic representation, in which all particles are free both before and after interaction. The obtained representations are used to construct the correct asymptotics of the partial components of the wave function of $N$ particles in the hyperspherical representation.