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 ^2x_2^2 $, where $0<\Lambda \not =1$ and $x=(x_1, x_2)\in R^2$. For any fixed rotational velocity $0\le \Omega <\Omega ^*:=2\min \{1, \Lambda\}$, it is known that ground states exist if and only if $ a<a^*$ for some critical constant $0<a^*<\infty$, 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 $a\nearrow a^*$, 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 $R(a):=\{x\in R^2:\,|x|\le C (a^*-a)^{-\frac{1}{12}}\}$ as $a\nearrow a^*$ for some constant $C>0$, which is independent of $0<a<a^*$.