On the Cauchy problem for the Hartree approximation in quantum dynamics

We prove existence and uniqueness results for the time-dependent Hartree approximation arising in quantum dynamics. The equations of motion form a coupled system nonlinear Schr{\"o}dinger evolution product state approximations. They are prominent example dimension reduction context Dirac-Frenkel variational principle. handle case Coulomb potentials thanks to Strichartz estimates. Our main result addresses general setting where coupling cannot be considered as perturbation. proof uses recursive construction that is inspired by standard approach Cauchy problem associated symmetric quasilinear hyperbolic equations.