Ground states of attractive Bose gases near the critical rotating velocity

Abstract

We study ground states of attractive Bose gases, which are confined in a harmonic trap $V(x)=x_1^2+\Lambda x_2^2$ ($\Lambda \ge 1$) rotating at the velocity $\Omega$. For any $0\le \Omega <\Omega ^*:=2$, where $\Omega ^*$ is called a critical rotational velocity, it is well 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. In this paper, we consider the critical rotating case, where the rotational velocity $\Omega =\Omega ^*$, to study the existence and non-existence of ground states with respect to $a>0$. As imposed in Remark 2.2 of Lewin et al. (Blow-up profile of rotating 2D focusing bose gases. macroscopic limits of quantum systems, Springer, Berlin, 2018), we also analyze the limiting behavior of ground states as $a\nearrow a^*$ for the case where $\Omega =\Omega _{a}:=\Omega ^*\sqrt{1-C_0(a^*-a)^m}\nearrow \Omega ^*$, $0\le m<\frac{1}{2}$ and $0<C_0<1$.