Energy-critical Hartree equation with harmonic potential for radial data

In this paper, we consider the defocusing, energy-critical Hartree equation with harmonic potential for the radial data in all dimensions ( n ≥ 5 ) and show the global well-posedness and scattering theory in the space Σ = H 1 ∩ F H 1 . We take advantage of some symmetry of the Hartree nonlinearity to exploit the derivative-like properties of the Galilean operators and obtain the energy control as well. Based on Bourgain and Tao’s approach, we use a localized Morawetz identity to show the global well-posedness. A key decay estimate comes from the linear part of the energy rather than the nonlinear part, which finally helps us to complete the scattering theory.