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 $$ i \partial_t u +\Delta u + c|x|^{-2} u = -|u|^{\frac{4}{d}} u, \quad u(0) = u_0, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, $$ where $d\geq 3$ and $c\ne 0$ 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 $c<\lambda(d)$.