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

We consider the focusing nonlinear Schr"odinger 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.