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 $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u-c\mathrm{\mid }x{\mathrm{\mid }}^{-2}u=-\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u,u(0)=u_0\in H^1,(t,x)\in \mathbb{R}\times {\mathbb{R}}^d,$$ where $d\ge 3$, $\frac{4}{d}\le \alpha \le \frac{4}{d-2}$ and $c\mathrm{\ne }0$ 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 $d\ge 3$ 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 $d\ge 3$ 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 $d\ge 3$ and $c>-\frac{d^2+4d}{(d+2{)}^2}\lambda (d)$.