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

TL;DR

This paper studies the long-term behavior of dipolar Bose-Einstein condensates described by the Gross-Pitaevskii equation, proving scattering results below the ground state energy in both unstable and stable regimes.

Contribution

It establishes scattering for solutions below the ground state energy threshold and introduces a novel analysis combining variational methods, profile decomposition, and virial estimates.

Findings

Solutions below the ground state energy behave as free waves asymptotically.

In the stable regime, all initial data in the energy space scatter.

The paper provides a rigorous framework for understanding the asymptotic dynamics of dipolar quantum gases.

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.