Global dynamics below the ground states for NLS under partial harmonic confinement

We are concerned with the global behavior of the solutions of the focusing mass supercritical nonlinear Schr{"o}dinger equation under partial harmonic confinement. We establish a necessary and sufficient condition on the initial data below the ground states to determine the global behavior (blow-up/scattering) of the solution. Our proof of scattering is based on the varia-tional characterization of the ground states, localized virial estimates, linear profile decomposition and nonlinear profiles.