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Remarks on nonlinear Schrödinger equations with harmonic potential. (English) Zbl 1021.81013

This paper concerns the existence and blow up results for the Cauchy problem for a nonlinear Schrödinger equation with isotropic harmonic potential modeling the Bose-Einstein condensation. The local problem is treated as in the case with no potential. For the global problem an evolution law, which is the analogue of the pseudo-conformal conservation law for the nonlinear Schrödinger equation, is established. With this law, the author gives wave collapse criteria, as well as an upper bound for the blow up time. A lower time for the breaking time is also given.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81V70 Many-body theory; quantum Hall effect
35Q55 NLS equations (nonlinear Schrödinger equations)
82B26 Phase transitions (general) in equilibrium statistical mechanics