Uniqueness and stability of normalized ground states for Hartree equation with a harmonic potential

The dynamic properties of normalized ground states for the Hartree equation with a harmonic potential are addressed. The existence of normalized ground state for any prescribed mass is confirmed according to mass-energy constrained variational approach. The uniqueness is shown by the strictly convex properties of the energy functional. Moreover, the orbital stability of every normalized ground state is proven in terms of the Cazenave and Lions’ argument.