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 Ḣ¹/2-critical nonlinearity and with inverse-square potential(CNLSₐ) in dimensions d∈\3,4,5\ {equation}{NLS-ab} {cases} i∂_tu-L_au=-|u|4/d-2u+|u|4/d-1u, (t,x)^d,{CNLSₐ},\\ u(0,x)=u_0(x)∈ H^1_a(R^d), {cases} {equation} where Lₐ=-Δ+a|x|⁻² and the energy is below and equal to the threshold mₐ, which is given by the ground state Wₐ satisfying LₐWₐ=|Wₐ|4/d-2Wₐ. 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 {NLS-ab} to classify the dynamics of Hₐ¹-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.