Many-particle correlations in an $N$-particle system and many-particle scattering amplitudes

The expansion of the energy of the symmetric ground state of a system of $N$ particles with respect to the density parameter is studied. The method of stationary phase in the multi-dimensional case is used to obtain estimates with respect to the density parameter of the Feynman diagrams of the perturbation theory series. The connection between the coefficients in the expansion of the energy in powers of the density and the many-particle Faddeev scattering amplitudes is established.