Local Uniqueness of Ground States for Rotating Bose-Einstein Condensates with Attractive Interactions

TL;DR

This paper proves the local uniqueness of ground states for rotating Bose-Einstein condensates with attractive interactions near the critical coupling, extending previous results to more general traps without symmetry.

Contribution

It establishes the uniqueness of ground states near the critical coupling for a broad class of traps, including non-symmetric ones, under rotation.

Findings

Unique ground state exists up to a constant phase as coupling approaches critical value.

Result holds for general traps, not necessarily symmetric.

Extends previous symmetry-dependent uniqueness results.

Abstract

We study ground states of two-dimensional Bose-Einstein condensates with attractive interactions in a trap $V(x)$ rotating at the velocity $\Omega $. It is known that there exist a critical rotational velocity $0<\Omega ^*:=\Omega^*(V)\leq \infty$ and a critical number $0<a^*<\infty$ such that for any rotational velocity $0\le \Omega <\Omega ^*$, ground states exist if and only if the coupling constant $a$ satisfies $a<a^*$. For a general class of traps $V(x)$, which may not be symmetric, we prove in this paper that up to a constant phase, there exists a unique ground state as $a\nearrow a^*$, where $\Omega\in(0,\Omega^*)$ is fixed. This result extends essentially our recent uniqueness result, where only the radially symmetric traps $V(x)$ could be handled with.