Nonexistence of Vortices for Rotating Two-Component Focusing Bose Gases

TL;DR

This paper investigates the existence and nonexistence of vortices in ground states of rotating two-component Bose gases with attractive interactions, establishing conditions for their existence and proving vortices do not form near critical interaction strengths.

Contribution

It provides precise conditions for ground state existence and demonstrates the nonexistence of vortices as interspecies attraction approaches critical levels.

Findings

Ground states exist under specific interaction strength conditions.

Vortices do not form as interspecies attraction nears critical value.

Conditions for vortex nonexistence are established for fixed rotation and interaction parameters.

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 $0<\Omega <\Omega ^*:=2$, ground states exist if and only if $0<a_1,\, a_2<a^*:=\|w\|^2_2$ and $0<\beta<\beta^*:=a^*+\sqrt{(a^*-a_1)(a^*-a_2)}$, 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 $\beta\nearrow\beta^*$, where $0<\Omega <\Omega ^*$ and $0<a_1,\, a_2<a^*$ are fixed.