Ground States of Attractive Bose Gases in Rotating Anharmonic Traps

Abstract

This paper is concerned with ground states of attractive Bose gases confined in an anharmonic trap $V(x)=\omega (|x{|}^2+k|x{|}^4)$ rotating at the velocity $\Omega >0$, where $\omega >0$ denotes the trapping frequency, and $k>0$ represents the strength of the quartic term. It is known that for any $\Omega >0$, ground states exist in such traps if and only if $0<a<a^{\ast }$, where $a^{\ast }:=\Vert Q{\Vert }_2^2$ and $Q>0$ is the unique positive solution of $\Delta Q-Q+Q^3=0$ in ${\mathbb{R}}^2$. By analyzing the refined energies and expansions of ground states, we prove that there exists a constant $C>0$, independent of $0<a<a^{\ast }$, such that ground states do not have any vortex in the region $R(a):=\{x\in {\mathbb{R}}^2:|x|\le C(a^{\ast }-a{)}^{-\frac{1-6\beta }{20}}\}$ as $a\nearrow a^{\ast }$, for the case where $\omega =\frac{3{\Omega }^2}{4}$, $k=\frac{1}{6}$, and $\Omega =C_0(a^{\ast }-a{)}^{-\beta }$ varies for some $\beta \in [0,\frac{1}{6})$ and $C_0>0$.