The Nonexistence of Vortices for Rotating Bose-Einstein Condensates with Attractive Interactions

Abstract

This article is devoted to studying the model of two-dimensional attractive Bose-Einstein condensates in a trap $V(x)$ rotating at the velocity $\Omega$. This model can be described by the complex-valued Gross-Pitaevskii energy functional. It is shown that there exists a critical rotational velocity $0<{\Omega }^{\ast }:={\Omega }^{\ast }(V)\le \infty$, depending on the general trap $V(x)$, such that for any rotational velocity $0\le \Omega <{\Omega }^{\ast }$, minimizers (i.e., ground states) exist if and only if $a<a^{\ast }=\Vert w{\Vert }_2^2$, where $a>0$ denotes the absolute product for the number of particles times the scattering length, and $w>0$ is the unique positive solution of $\Delta w-w+w^3=0$ in ${\mathbb{R}}^2$. If $V(x)=|x{|}^2$ and $0<\Omega <{\Omega }^{\ast }(=2)$ is fixed, we prove that, up to a constant phase, all minimizers must be real-valued, unique and free of vortices as $a\nearrow a^{\ast }$, by analyzing the refined limit behavior of minimizers and employing the non-degenerancy of $w$.

arXiv comment41 pages, the current file is an updated version of arXiv:1901.09619 and the title is changed. This paper is accepted by Archive for Rational Mechanics and Analysis

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• Threshold for Blowup and Stability for Nonlinear Schrödinger Equation with Rotation
• The Nonexistence of Vortices for Rotating Bose-Einstein Condensates in Non-Radially Symmetric Traps
• Stabilization against collapse of 2D attractive Bose-Einstein condensates with repulsive, three-body interactions
• Multiplicity and asymptotics of standing waves for the energy critical half-wave
• Existence, non-existence and blow-up behavior of minimizers for the mass-critical fractional nonlinear Schrödinger equations with periodic potentials
• Multiplicity, asymptotics and stability of standing waves for nonlinear Schrödinger equation with rotation
• Ground states for 3D dipolar Bose-Einstein condensate involving quantum fluctuations and three-body losses
• Homogeneous attractive Bose-Einstein condensates with repulsive three-body interactions: the two-dimensional case
• Existence and Asymptotic Behavior of Ground States for Rotating Bose-Einstein Condensates