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We consider the focusing nonlinear Schr\"odinger equation with the criticalinverse square potential. We give the first proof of the uniqueness of theground state solution. Consequently, we obtain a sharpHardy-Gagliardo-Nirenberg interpolation inequality. Moreover, we provide acomplete characterization for the minimal mass blow-up solutions to the timedependent problem.

Uniqueness of ground state and minimal-mass blow-up solutions for focusing NLS with Hardy potential