On the dynamics of a quadratic Schr\"odinger system in dimension $n=5$

In this work we give a sharp criterion for the global well-posedness, in the energy space, for a system of nonlinear Schr"odinger equations with quadratic interaction in dimension $n=5$. The criterion is given in terms of the charge and energy of the ground states associated with the system, which are obtained by minimizing a Weinstein-type functional. The main result is then obtained in view of a sharp Gagliardo-Nirenberg-type inequality.