We consider the focusing nonlinear Schrödinger equation with the critical inverse square potential. We give the first proof of the uniqueness of the ground state solution. Consequently, we obtain a sharp Hardy-Gagliardo-Nirenberg interpolation inequality. Moreover, we provide a complete characterization for the minimal mass blow-up solutions to the time dependent problem.

中文翻译：

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基态的唯一性和最小质量的爆破解决方案，用于聚焦具有Hardy势的NLS

我们考虑具有临界反平方电势的聚焦非线性Schrödinger方程。我们给出了基态解唯一性的第一个证明。因此，我们得到了一个尖锐的Hardy-Gagliardo-Nirenberg插值不等式。此外，我们提供了针对时间相关问题的最小质量爆炸解决方案的完整表征。