SCATTERING OF THE THREE-DIMENSIONAL CUBIC NONLINEAR SCHRODINGER EQUATION WITH PARTIAL HARMONIC POTENTIALS

We consider the following three-dimensional defocusing cubic nonlinear Schrodinger equation (NLS) with partial harmonic potential: c a atu C (A3 - x 2 )u D j u j 2 u ; (NLS) u j t = 0 D u0 center dot Our main result shows that the solution u scatters for any given initial data u0 with finite mass and energy. The main new ingredient in our approach is to approximate (NLS) in the large-scale case by a relevant dispersive continuous resonant (DCR) system. The proof of global well-posedness and scattering of the new (DCR) system is greatly inspired by the fundamental works of Dodson (2012, 2016) in his study of scattering for the mass-critical nonlinear Schrodinger equation. The analysis of (DCR) system allows us to utilize the additional regularity of the smooth nonlinear profile so that the celebrated concentrationcompactness/rigidity argument of Kenig and Merle (2006, 2008) applies.