Nonexistence of Vortices for Rotating Two-Component Focusing Bose Gases

Abstract

This paper is concerned with ground states of two-component Bose gases confined in a harmonic trap $V(x)=x_1^2+{\Lambda }^2{x}_2^2$ rotating at the velocity $\Omega >0$, where $\Lambda \ge 1$ and $(x_1,x_2)\in R^2$. We focus on the case where the intraspecies interaction $(-a_1,-a_2)$ and the interspecies interaction $-\beta$ are both attractive, i.e, $a_1,a_2$ and $\beta$ are all positive. It is shown that for any $0<\Omega <{\Omega }^{\ast }:=2$, ground states exist if and only if $0<a_1,a_2<a^{\ast }:=\Vert w{\Vert }_2^2$ and $0<\beta <{\beta }^{\ast }:=a^{\ast }+\sqrt{(a^{\ast }-a_1)(a^{\ast }-a_2)}$, where $w>0$ is the unique positive solution of $-\Delta w+w-w^3=0$ in $R^2$. By developing the argument of refined expansions, we further prove the nonexistence of vortices for ground states as $\beta \nearrow {\beta }^{\ast }$, where $0<\Omega <{\Omega }^{\ast }$ and $0<a_1,a_2<a^{\ast }$ are fixed.

59 pages, 1 figure, and all comments are welcome