The Cauchy Problem of the Hartree Equation

In this paper, we systematically study the wellposedness, illposedness of the Hartree equation, and obtain the sharp local wellposedness, the global existence in H s , s ≥ 1 and the small scattering result in H s for 2 < γ < n and s ≥ 2 − 1. In addition, we study the nonexistence of nontrivial asymptotically free solutions of the Hartree equation.