Threshold Scattering for the Energy-Critical NLS with a Repulsive Inverse Square Potential

We study the threshold scattering problem for the energy-critical nonlinear Schrödinger equation with a repulsive inverse-square potential $\frac{a}{\mathrm{\mid }x{\mathrm{\mid }}^2}>0$ in dimensions $d=4,5,6$. On the energy level surface determined by the ground state of the energy-critical NLS without potential, we show that, despite the absence of a ground state in this setting, a strong form of rigidity persists below the kinetic threshold. Specifically, we prove that any solution on this energy surface with kinetic energy strictly below that of the ground state is global and scatters to zero. Our approach combines refined modulation analysis, a center-translated global Virial estimate, and a bootstrap argument to control the modulation parameters.