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 $w$.

41 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