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.

中文翻译：

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关于维度 $n = 5$ 的二次薛定谔系统的动力学

在这项工作中，我们为能量空间中的全局适定性给出了一个尖锐的标准，对于在维度 $n=5$ 上具有二次相互作用的非线性 Schr\"odinger 方程组。该标准是根据电荷给出的和系统相关的基态能量，它们是通过最小化 Weinstein 型泛函获得的。然后根据尖锐的 Gagliardo-Nirenberg 型不等式获得主要结果。