Dynamics of the combined nonlinear Schr\"odinger equation with inverse-square potential

We consider the long-time dynamics of focusing energy-critical Schr\"odinger equation perturbed by the $\dot{H}^\frac{1}{2}$-critical nonlinearity and with inverse-square potential(CNLS$_a$) in dimensions $d\in\{3,4,5\}$ \begin{equation}\label{NLS-ab} \begin{cases} i\partial_tu-\mathcal{L}_au=-|u|^{\frac{4}{d-2}}u+|u|^{\frac{4}{d-1}}u, \quad (t,x)\in\mathbb{R}\times\mathbb{R}^d,\tag{CNLS$_a$},\\ u(0,x)=u_0(x)\in H^1_a(\mathbb{R}^d), \end{cases} \end{equation} where $\mathcal{L}_a=-\Delta+a|x|^{-2}$ and the energy is below and equal to the threshold $m_a$, which is given by the ground state $W_a$ satisfying $\mathcal{L}_aW_a=|W_a|^{\frac{4}{d-2}}W_a$. When the energy is below the threshold, we utilize the concentration-compactness argument as well as the variatonal analysis to characterize the scattering and blow-up region. When the energy is equal to the threshold, we use the modulation analysis associated to the equation \eqref{NLS-ab} to classify the dynamics of $H_a^1$-solution. In both regimes of scattering results, we do not need the radial assumption in $d=4,5$. Our result generalizes the scattering results of [31-33] and [3] in the setting of standard combined NLS.