On the Cauchy problem for the Hartree approximation in quantum dynamics

Abstract

We prove existence and uniqueness results for the time-dependent Hartree approximation arising in quantum dynamics. The Hartree equations of motion form a coupled system of nonlinear Schrödinger equations for the evolution of product state approximations. They are a prominent example for dimension reduction in the context of the time-dependent Dirac–Frenkel variational principle. Our main result addresses a general setting with smooth potentials where the nonlinear coupling cannot be considered as a perturbation. The proof uses a recursive construction that is inspired by the standard approach for the Cauchy problem associated to symmetric quasilinear hyperbolic equations. We also discuss the case of Coulomb potentials, though treated differently (using Strichartz estimates and a classical fixed point argument).