This note investigates the coupled Schrödinger system

$$\begin{aligned} i\dot{u}_j +\Delta u_j= -\left( \sum \nolimits _{k=1}^{m}a_{jk}|u_k|^p\right) |u_j|^{p-2}u_j. \end{aligned}$$

Indeed, beyond the mass-energy threshold given in Saanouni (Appl Anal, 2020. https://doi.org/10.1080/00036811.2020.1808201), a scattering versus finite time blow-up dichotomy is obtained in the mass super-critical and energy sub-critical regime. Moreover, one extends the previous work [18] to the non-radial case.

中文翻译：

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重新审视聚焦耦合薛定谔系统的散射阈值

本笔记研究了耦合薛定谔系统

$$\begin{aligned} i\dot{u}_j +\Delta u_j= -\left( \sum \nolimits _{k=1}^{m}a_{jk}|u_k|^p\right) | u_j|^{p-2}u_j。\end{对齐}$$

事实上，除了 Saanouni (Appl Anal, 2020. https://doi.org/10.1080/00036811.2020.1808201) 中给出的质量-能量阈值之外，在质量超临界和能源亚临界状态。此外，将之前的工作 [18] 扩展到非径向情况。