We first introduce the basic ingredients of the eigenfunctional theory, and show that a D-dimensional quantum many-particle system is mapped into a (D+1)-dimensional time-depending single-particle problem, and in the representation of the eigenfunctionals of the particle propagator, the particles become free. Then using this method, we study five kinds of quantum many-particle systems: interacting boson system, repulsive, attractive interacting fermion systems, Hubbard model and single-impurity scattering in one-dimensional fermion system, and demonstrate that the microscopic Bogoliubov theory and the phenomenological Bijl–Feynman theory of the bosons are closely related, and apart from an anti-symmetry factor Det ‖eikj·xl‖ the ground state wave function of the repulsive interacting fermion system has a similar form to that of the interacting boson system. Moreover, we show that the attractive interacting fermion system has a sound-type excitation spectrum like that in the interacting boson system. For one-dimensional Hubbard model we calculate the electron Green's function, and charge and spin density–density correlation functions which are consistent with the exact ones obtained by the Bethe ansatz and numerical calculations, and show that the ground state energy is increasing with U, and the electrons has single-occupied constraint in the large U limit. Finally, we demonstrate clearly the evolution of the system from its ultraviolet fixed point to infrared critical fixed point as the impurity potential increases. At the infrared critical fixed point, the fermion Green's function shows that the fermions are completely reflected on the impurity site.
