Abstract. We revisit the scattering result of Holmer and Roudenko [5] on the radial focusing cubic NLS in three space dimensions. Using the radial Sobolev embedding and a virial/Morawetz estimate, we give a simple proof of scattering below the ground state that avoids the use of concentration compactness.

We study the nonlinear Schrödinger equation with an inversesquare potential in dimensions 3 ≤ d ≤ 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.

We revisit the scattering result of Duyckaerts, Holmer, and Roudenko for the non-radialḢ 1/2 -critical focusing NLS. By proving an interaction Morawetz inequality, we give a simple proof of scattering below the ground state in dimensions d ≥ 3 that avoids the use of concentration compactness.

We consider a class of power-type nonlinear Schrödinger equations for which the power of the nonlinearity lies between the mass-and energy-critical exponents. Following the concentration-compactness approach, we prove that if a solution u is bounded in the critical Sobolev space throughout its lifespan, that is, u ∈ L ∞ tḢ sc x , then u is global and scatters.

We consider a class of defocusing energy-supercritical nonlinear Schrödinger equations in four space dimensions. Following a concentrationcompactness approach, we show that for 1 < sc < 3/2, any solution that remains bounded in the critical Sobolev spaceḢ sc x (R 4 ) must be global and scatter. Key ingredients in the proof include a long-time Strichartz estimate and a frequency-localized interaction Morawetz inequality.