Faddeev equations and the time independent mean field theory

To obtain the solutions of the quantum mechanic equations, the formalism of the Schrodinger equation is insufficient for a few body system. In fact, it's, for example, impossible to describe with just one equation the physic of the problem. In the collision theory, the Lippmann-Schwinger equation is fundamental. But, it's still difficult to have the exact solution when we study a system of N particles with N>2. In this note, we first study, by the simplest form of Faddeev equations, the elastic scattering of a particle 1 on a bound state of two particles 2 and 3. We want to calculate the three-body Faddeev amplitude from a variational principle. We find a functional whose stationary conditions are Faddeev equations. A mean-field approximation solutions can be introduced in this variational principle, in order to generate simplified approximate solutions of Faddeev equations and fast estimates of the three-body and more collisions amplitudes