Asymptotic dynamic for Dipolar Quantum Gases below the ground state energy threshold

We consider the Gross-Pitaevskii equation describing a dipolar Bose-Einstein condensate without external confinement. We first consider the unstable regime, where the nonlocal nonlinearity is neither positive nor radially symmetric and standing states are known to exist. We prove that under the ener...

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Bibliographic Details
Published in:Journal of functional analysis 2019-09, Vol.277 (6), p.1958-1998
Main Authors: Bellazzini, Jacopo, Forcella, Luigi
Format: Article
Language:English
Subjects:
Asymptotic behavior
BEC
Gross-Pitaevskii equation
NLS-like equation
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Summary:We consider the Gross-Pitaevskii equation describing a dipolar Bose-Einstein condensate without external confinement. We first consider the unstable regime, where the nonlocal nonlinearity is neither positive nor radially symmetric and standing states are known to exist. We prove that under the energy threshold given by the ground state, all global in time solutions behave as free waves asymptotically in time. The ingredients of the proof are variational characterization of the ground states energy, a suitable profile decomposition theorem and localized virial estimates, enabling to carry out a Concentration/Compactness and Rigidity scheme. As a byproduct we show that in the stable regime, where standing states do not exist, any initial data in the energy space scatters.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2019.04.005