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 Ḣ¹2-critical nonlinearity and with inverse-square potential (CNLSₐ) in dimensions d\3, 4, 5\ equationNLS-ab cases iₜu-Lₐu=-|u|^4{d-2}u+|u|^4{d-1}u, (t, x) ᵈ, CNLSₐ, \\ u (0, x) =u₀ (x) H¹ₐ (Rᵈ), cases equation where Lₐ=-+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-2}Wₐ. 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.