Global existence and blowup for a class of the focusing nonlinear Schr\"odinger equation with inverse-square potential

We consider a class of the focusing nonlinear Schr\"odinger equation with inverse-square potential \[ i\partial_t u + \Delta u -c|x|^{-2}u = - |u|^\alpha u, \quad u(0)=u_0 \in H^1, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d, \] where $d\geq 3$, $\frac{4}{d}\leq \alpha \leq \frac{4}{d-2}$ and $c\ne 0$ satisfies $c>-\lambda(d):=-\left(\frac{d-2}{2}\right)^2$. In the mass-critical case $\alpha=\frac{4}{d}$, we prove the global existence and blowup below ground states for the equation with $d\geq 3$ and $c>-\lambda(d)$. In the mass and energy intercritical case $\frac{4}{d}<\alpha<\frac{4}{d-2}$, we prove the global existence and blowup below the ground state threshold for the equation. This extends similar results of \cite{KillipMurphyVisanZheng} and \cite{LuMiaoMurphy} to any dimensions $d\geq 3$ and a full range $c>-\lambda(d)$. We finally prove the blowup below ground states for the equation in the energy-critical case $\alpha=\frac{4}{d-2}$ with $d\geq 3$ and $c>-\frac{d^2+4d}{(d+2)^2} \lambda(d)$.