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Stability and instability properties of rotating Bose-Einstein condensates

Jack Arbunich, Irina Nenciu, Christof Sparber

Abstract

We consider the mean-field dynamics of Bose-Einstein condensates in rotating harmonic traps and establish several stability and instability properties for the corresponding solution. We particularly emphasize the difference between the situation in which the trap is symmetric with respect to the rotation axis and the one where this is not the case.

Stability and instability properties of rotating Bose-Einstein condensates

Abstract

We consider the mean-field dynamics of Bose-Einstein condensates in rotating harmonic traps and establish several stability and instability properties for the corresponding solution. We particularly emphasize the difference between the situation in which the trap is symmetric with respect to the rotation axis and the one where this is not the case.

Paper Structure

This paper contains 4 sections, 10 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Existence of ground states
  3. Orbital stability
  4. A resonance-type phenomenon in non-isotropic potentials

Key Result

Proposition 2.1

Let $|\Omega|< \omega$ and Assumption hyp1 hold. Then for any $\varphi\in \Sigma$ with $\|\varphi\|_{L^{2}}^{2}=N$, there is a $\delta>0$ such that Moreover, $\varphi\mapsto{E}_{\Omega}(\varphi)$ is weakly lower semicontinuous in $\Sigma$, i.e. for $\{ \varphi_{k} \}^{\infty}_{k=1} \subset \Sigma$ such that $\varphi_{k} \rightharpoonup\varphi \in \Sigma$, we have

Theorems & Definitions (22)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Theorem 3.1: Orbital stability of ground states
  • proof
  • Remark 3.2
  • Corollary 3.3
  • ...and 12 more