Mass concentration and characterization of finite time blow-up solutions for the nonlinear Schr\"odinger equation with inverse-square potential

We consider the $L^2$-critical NLS with inverse-square potential $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u+c\mathrm{\mid }x{\mathrm{\mid }}^{-2}u=-\mathrm{\mid }u{\mathrm{\mid }}^{\frac{4}{d}}u,u(0)=u_0,(t,x)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^d,$$ where $d\ge 3$ and $c\mathrm{\ne }0$ satisfies $c<\lambda(d) := \left(\frac{d-2}{2}\right)^2$. Using a refined compactness lemma, we extend the mass concentration of finite time blow-up solutions established in the attractive case by the first author in [Bensouilah] to $c<\lambda (d)$. By means of a simple and short limiting profile theorem, we get the same classification result obtained by Csobo and Genoud in [CsoboGenoud] for $0<c<\lambda (d)$. It also enables us to extend the classification to $c<\lambda (d)$.