Scattering in $H^1$ for the intercritical NLS with an inverse-square potential

We study the nonlinear Schr"odinger equation with an inverse-square potential in dimensions $3\le d\le 6$. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing case, we prove a scattering/blowup dichotomy below the ground state. In the defocusing case, we prove scattering in $H^1$ for arbitrary data.