The energy-critical NLS with inverse-square potential

Abstract

We consider the defocusing energy-critical nonlinear Schrödinger equation with inverse-square potential $iu_t = -Δ u + a|x|^{-2}u + |u|^4u$ in three space dimensions. We prove global well-posedness and scattering for $a > - \frac{1}{4} + \frac{1}{{25}}$. We also carry out the variational analysis needed to treat the focusing case.