The Role of dimensionality in the stability of a confined Bose gas

We study analytically the ground-state stability of a Bose-Einstein condensate (BEC) confined in an harmonic trap with repulsive or attractive zero-range interaction by minimizing the energy functional of the system. In the case of repulsive interaction the BEC mean radius grows by increasing the number of bosons, instead in the case of attractive interaction the BEC mean radius decreases by increasing the number of bosons: to zero if the system is one-dimensional and to a minimum radius, with a maximum number of bosons, if the system is three-dimensional.