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Mathematics > Analysis of PDEs

arXiv:1808.01626 (math)
[Submitted on 5 Aug 2018 (v1), last revised 6 May 2019 (this version, v2)]

Title:Asymptotic dynamic for dipolar Quantum Gases below the ground state energy threshold

Authors:Jacopo Bellazzini, Luigi Forcella
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Abstract: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.
Comments: Fixed typos and minor changes. Journal of Functional Analysis, in press
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
Cite as: arXiv:1808.01626 [math.AP]
  (or arXiv:1808.01626v2 [math.AP] for this version)
  https://doi.org/10.48550/arXiv.1808.01626
Related DOI: https://doi.org/10.1016/j.jfa.2019.04.005

Submission history

From: Luigi Forcella [view email]
[v1] Sun, 5 Aug 2018 14:21:47 UTC (30 KB)
[v2] Mon, 6 May 2019 14:24:59 UTC (31 KB)
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