In this paper, we study the solutions to the energy-critical quadratic nonlinear Schrodinger system in ${dot H}^1times{dot H}^1$, where the sign of its potential energy can not be determined directly. If the initial data ${rm u}_0$ is radial or non-radial but satisfies the mass-resonance condition, and its energy is below that of the ground state, using the compactness/rigidity method, we give a complete classification of scattering versus blowing-up dichotomies depending on whether the kinetic energy of ${rm u}_0$ is below or above that of the ground state.
In this paper, we simplify the proof of M. Hamano in cite{Hamano2018}, scattering theory of the solution to eqref{NLS system}, by using the method from B. Dodson and J. Murphy in cite{Dodson2018}. Firstly, we establish a criterion to ensure the solution scatters in $ H^1(mathbb{R}^5) times H^1(mathbb{R}^5) $. In order to verify the correctness of the condition in scattering criterion, we must exclude the concentration of mass near the origin. The interaction Morawetz estimate and Galilean transform characterize a decay estimate, which implies that the mass of the system cannot be concentrated.
