The Nonexistence of Vortices for Rotating Bose-Einstein Condensates in Non-Radially Symmetric Traps

Abstract

We consider ground states of rotating Bose-Einstein condensates with attractive interactions in non-radially harmonic traps $V(x)=x_1^2+{\Lambda }^2{x}_2^2$, where $0<\Lambda \ne 1$ and $x=(x_1,x_2)\in R^2$. For any fixed rotational velocity $0\le \Omega <{\Omega }^{\ast }:=2\min \{1,\Lambda \}$, it is known that ground states exist if and only if $a<a^{\ast }$ for some critical constant $0<a^{\ast }<\infty$, where $a>0$ denotes the product for the number of particles times the absolute value of the scattering length. We analyze the asymptotic expansions of ground states as $a\nearrow a^{\ast }$, which display the visible effect of $\Omega$ on ground states. As a byproduct, we further prove that ground states do not have any vortex in the region $R(a):=\{x\in R^2:|x|\le C(a^{\ast }-a{)}^{-\frac{1}{12}}\}$ as $a\nearrow a^{\ast }$ for some constant $C>0$, which is independent of $0<a<a^{\ast }$.

arXiv comment29 pages, this is the revised version. The further revised version is published by Journal de Mathématiques Pures et Appliquées in 2023

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