Scattering of the focusing energy-critical NLS with inverse square potential

Abstract

We consider the Cauchy problem for the focusing energy-critical nonlinear Schrödinger equation with an inverse square potential in dimension $ d = 4, 5, 6 $. We show that if the supremum of the kinetic energy of a solution over its maximal lifespan is less than the kinetic energy of the ground state, then the solution must exist globally in time and scatter in both time directions. We develop the 'long-term kinetic energy decoupling' associated with the appearance of inverse square potential. No radial assumption is made on the initial data. This extends the result in [22] by the first author to the non-radial case.