Dynamics of the combined nonlinear Schrödinger equation with inverse-square potential

We consider the long-time dynamics of focusing energy-critical Schrödinger 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 ${\mathcal{L}}_a=-\mathrm{\Delta }+a\mathrm{\mid }x{\mathrm{\mid }}^{-2}$ and the energy is below and equal to the threshold $m_a$, which is given by the ground state $W_a$ satisfying ${\mathcal{L}}_a{W}_a=\mathrm{\mid }{W}_a{\mathrm{\mid }}^{\frac{4}{d-2}}{W}_a$. 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.