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

TL;DR

This paper investigates the behavior of ground states in rotating Bose-Einstein condensates with attractive interactions in non-radially symmetric traps, showing the absence of vortices in certain regions as the system approaches critical interaction strength.

Contribution

It provides asymptotic analysis of ground states near criticality and proves the nonexistence of vortices in specific regions for these states.

Findings

Ground states exist only if the interaction parameter is below a critical value.

Asymptotic expansions reveal the influence of rotation on ground states near criticality.

No vortices are present in a specified region as the system approaches the critical interaction strength.

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 $0<a<a^*$.