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In this article, we consider the dynamics of the energy-critical quadraticnonlinear Schr\"odinger system in energy-space $ {\dot H}^1 \times {\dotH}^1\times{\dot H}^1 $. Under this circumstance, we don't know whether thepotential energy is positive or negative. In addition, we discover a family ofnew physically conserved quantity under mass-resonance, which is an interestingfact in this article. Using it and by concentration compactness/rigidityarguement, we will prove the scattering for this system if the energy ofinitial data is below the energy of ground state and the system satisfies themass-resonance. Besides, we also study the blow-up of this system, when theenergy of initial data is above the energy of ground state.

Scattering and blow-up for the energy-critical quadratic nonlinear Schr\"odinger system with three wave interaction