Global existence of solutions to the Cauchy problem for time-dependent Hartree equations

The existence of global solutions to the Cauchy problem for time-dependent Hartree equations for N electrons is established. The solution is shown to have a uniformly bounded H1(R3) norm and to satisfy an estimate of the form ∥ ψ (t) ∥H2 ⩽ c exp(kt). It is shown that ’’negative energy’’ solutions do not converge uniformly to zero as t → ∞.