Fluctuations around Hartree states in the mean-field regime

We consider the dynamics of a large system of N interacting bosons in the mean-field regime where the interaction is of order 1/N. We prove that the fluctuations around the nonlinear Hartree state are generated by an effective quadratic Hamiltonian in Fock space, which is derived from Bogoliubov's approximation. We use a direct method in the N-particle space, which is different from the one based on coherent states in Fock space.