Sharp threshold of global existence and instability of standing wave for the Schrödinger-Hartree equation with a harmonic potential

In this paper, we consider the Schrödinger-Hartree equation with a harmonic potential. By constructing some cross-invariant manifolds of the evolution flow and some variational problems, we obtain the sharp threshold for global existence and blow-up of the solutions. In addition, we discuss the stability and instability of the standing waves. In particular, we give two different characterizations on these problems in the L2 critical case. Our results extend some earlier results.