The energy-critical NLS with inverse-square potential

We consider the defocusing energy-critical nonlinear Schr"odinger equation with inverse-square potential $i{u}_t=-\mathrm{\Delta }u+a\mathrm{\mid }x{\mathrm{\mid }}^{-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{4}u$ in three space dimensions. We prove global well-posedness and scattering for $a>-\frac14 +\frac1{25}$. We also carry out the variational analysis needed to treat the focusing case.