Uniqueness and orbital stability of standing waves for the nonlinear schro"\dinger equation with a partial confinement

We consider the 3d cubic nonlinear Schro"\dinger equation (NLS) with a strong 2d harmonic potential. The model is physically relevant to observe the lower -dimensional dynamics of the Bose-Einstein condensate, but its ground state cannot be constructed by the standard method due to its supercritical nature. In Bellazzini et al. [Comm. Math. Phys. 353 (2017), pp. 229--251], a proper ground state is constructed, introducing a constrained energy minimization problem. In this paper, we further investigate the properties of the ground state. First, we show that, as the partial confinement is increased, the 1d ground state is derived from the 3d energy minimizer with a precise rate of convergence. Then, by employing this dimension reduction limit, we prove the uniqueness of the 3d minimizer, provided that the confinement is sufficiently strong. Consequently, we obtain the orbital stability of the minimizer, which improves that of the set of minimizers in the previous work Bellazini et al. [Comm. Math. Phys., 353 (2017), pp. 229--251].