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We consider the 3d cubic nonlinear Schr\"odinger equation (NLS) with a strong2d harmonic potential. The model is physically relevant to observe thelower-dimensional dynamics of the Bose-Einstein condensate, but its groundstate cannot be constructed by the standard method due to its supercriticalnature. In Bellazzini-Boussa\"id-Jeanjean-Visciglia \cite{BBJV}, the authorsconstructed a proper ground state introducing a constrained energy minimizationproblem. In this paper, we further investigate the properties of the groundstate. First, we show that as the partial confinement is increased, the 1dground state is derived from the 3d energy minimizer with a precise rate ofconvergence. Then, by employing this dimension reduction limit, we prove theuniqueness of the 3d minimizer provided that the confinement is sufficientlystrong. Consequently, we obtain the orbital stability of the minimizer, whichimproves that of the set of minimizers in the previous work \cite{BBJV}.

Uniqueness and orbital stability of standing waves for the nonlinear Schrodinger equation with a partial confinement