Scattering of the three-dimensional cubic nonlinear Schrodinger equation with partial harmonic potentials

In this paper, we consider the following three dimensional defocusing cubic nonlinear Schrodinger equation (NLS) with partial harmonic potential begin{equation*}tag{NLS} ipartial_t u + left(Delta_{mathbb{R}^3 }-x^2 right) u = |u|^2 u, quad u|_{t=0} = u_0. end{equation*} Our main result shows that the solution $u$ scatters for any given initial data $u_0$ 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 cite{D3,D1,D2} 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 concentration-compactness/rigidity argument of Kenig and Merle applies.