Boundedness of energy for N-body Schrödinger equations with time dependent small potentials

We prove that Sobolev norms of solutions to time dependent Schrödinger equations for $d$-dimensional $N$-partcles interacting via time dependent two body potentials are bounded in time if certain Lebesgue norms of the potentials are small uniformly in time. The proof uses the scattering theory in the extended phase space which proves that all particles scatter freely in the remote past and far future.