Stability of attractive Bose-Einstein condensates

We propose the critical nonlinear Schrodinger equation with a harmonic potential as a model of attractive Bose-Einstein condensates. By an elaborate mathematical analysis we show that a sharp stability threshold exists with respect to the number of condensate particles. The value of the threshold agrees with the existing experimental data. Moreover with this threshold we prove that a ground state of the condensate exists and is orbital stable. We also evaluate the minimum of the condensate energy.