  We consider the dynamics of a large number N of nonrelativistic bosons in the
mean field limit for a class of interaction potentials that includes Coulomb
interaction. In order to describe the fluctuations around the mean field
Hartree state, we introduce an auxiliary Hamiltonian on the N-particle space
that is very similar to the one obtained from Bogoliubov theory. We show
convergence of the auxiliary time evolution to the fully interacting dynamics
in the norm of the N-particle space. This result allows us to prove several
other results: convergence of reduced density matrices in trace norm with
optimal rate, convergence in energy trace norm, and convergence to a time
evolution obtained from the Bogoliubov Hamiltonian on Fock space with expected
optimal rate. We thus extend and quantify several previous results, e.g., by
providing the physically important convergence rates, including time-dependent
external fields and singular interactions, and allowing for general initial
states, e.g., those that are expected to be ground states of interacting
systems.
