Ground states for dipolar quantum gases in the unstable regime

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  • 17 October 2014
Abstract
We study the existence and stability/ instability properties of standing waves for a nonlinear Schr\"odinger equation arising in dipolar Bose-Einstein condensate in the unstable regime. Two cases are studied: the first when the system is free, the second when gradually a trapping potential is add. In both cases this leads to the search of critical points of a constrained functional which is unbounded from below on the constraint. In the first case we prove the existence and instability of ground states by showing that the constrained functional has a so-called {\it mountain pass geometry}. In the second case we prove that, if the trapping potential is small, two different kind of standing waves appears: one corresponds to a local minima of the constrained energy functional, the other is again of {\it mountain pass type}. Despite the problem is mass supercritical and the functional unbounded from below the standing waves associated to the local minima are orbitally stable. Eventually we show that the addition of the trapping potential, however small, create a "gap" in the ground state energy level of the system.

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