Stability of the D-dimensional nonlinear Schrodinger equation under confined potential

We investigate the stability of the wavefunction of the D-dimensional nonlinear Schrodinger equation with harmonic potential terms. For the repulsive case, we prove that the wavefunction is absolutely stable. For the attractive case, by extending the Zakharov's theory, we show the emergence of the singularity of the wavefunction, termed the collapse, in a finite time. The conditions for the collapse include an extension of Weinstein's for the "free" nonlinear Schrodinger field. Further, me estimate an upper bound of the collapse-time where the collapse occurs without oscillations.