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We show that if a solution of the defocusing cubic NLS in 3d remains bounded in the homogeneous Sobolev norm of order 1/2 in its maximal interval of existence, then the interval is infinite and the solution scatters. No radial assumption is made.

Scattering for 𝐻̇^{1/2} bounded solutions to the cubic, defocusing NLS in 3 dimensions