Stability of the<i><i>D</i></i>-Dimensional Nonlinear Schrödinger Equation under Confined Potential

We investigate the stability of the wavefunction of the D -dimensional nonlinear Schrödinger 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 Schrödinger field. Further, we estimate an upper bound of the collapse-time where the collapse occurs without oscillations.

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