On stability and instability of standing waves for the nonlinear Schr\"odinger equation with inverse-square potential

We consider 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 $d \geq 3$, $c\ne 0$, $c<\lambda(d)=\left(\frac{d-2}{2}\right)^2$ and $0<\alpha\leq \frac{4}{d}$. Using the profile decomposition obtained recently by the first author \cite{Bensouilah}, we show that in the $L^2$-subcritical case, i.e. $0<\alpha<\frac{4}{d}$, the sets of ground state standing waves are orbitally stable. In the $L^2$-critical case, i.e. $\alpha=\frac{4}{d}$, we show that ground state standing waves are strongly unstable by blow-up.