  We prove that the Schr\"odinger equation for N number of particles in the
time dependent electro-magnetic field generates a unique unitary propagator on
the state space under the condition that the field is smooth and moderately but
almost critically increases at the spatial infinity such that propagator for
every single particle in the field enjoys the time local Strichartz estimates
and that the time dependent inter-particle potentials are almost critically
singular for Hamiltonians to have a unique selfadjoint realization at every
fixed time. We also show that the domain of definition of the quantum harmonic
oscillator is invariant under the propagator and, for initial states in that
space, solutions are continuously differentiable function of time variable with
values in the state space under the additional assumption that the time
derivative of inter-particle potentials exists almost everywhere and it
increases the spatial singularities by at most the inverse power of the
inter-particle distances. New estimates of Strichartz type for the propagator
for N independent particles in the field are proved and used for the proof.
