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

In this paper, we consider the following three dimensional defocusing cubic nonlinear Schrödinger equation (NLS) with partial harmonic potential
$\begin{equation}
\left\{\begin{array}{l}
i\partial_tu + \left(\Delta_{\mathbb{R}^3}-x^2\right)u = |u|^2u, \\
u|_{t=0} = u_0 \\
\end{array}\right. \tag{NLS}
\end{equation}$
Out 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 approxmate (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 [29, 31, 32] in his study of scattering for the mass-critical nonlinear Schrödinger 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 [61,62] applies.