This paper investigates the numerical approximation of ground states of rotating Bose-Einstein condensates. This problem requires the minimization of the Gross-Pitaevskii energy $E$ on a Riemannian manifold $\mathbb{S}$. To find a corresponding minimizer $u$, we use a generalized Riemannian gradient method that is based on the concept of Sobolev gradients in combination with an adaptively changing metric on the manifold. By a suitable choice of the metric, global energy dissipation for the arising gradient method can be proved. The energy dissipation property in turn implies global convergence to the density $|u|^2$ of a critical point $u$ of $E$ on $\mathbb{S}$. Furthermore, we present a precise characterization of the local convergence rates in a neighborhood of each ground state $u$ and how these rates depend on the first spectral gap of $E^{\prime\prime}(u)$ restricted to the $L^2$-orthogonal complement of $u$. With this we establish the first convergence results for a Riemannian gradient method to minimize the Gross-Pitaevskii energy functional in a rotating frame. At the same, we refine previous results obtained in the case without rotation. The major complication in our new analysis is the missing isolation of minimizers, which are at most unique up to complex phase shifts. For that, we introduce an auxiliary iteration in the tangent space $T_{\mathrm{i} u} \mathbb{S}$ and apply the Ostrowski theorem to characterize the asymptotic convergence rates through a weighted eigenvalue problem. Afterwards, we link the auxiliary iteration to the original Riemannian gradient method and bound the spectrum of the weighted eigenvalue problem to obtain quantitative convergence rates. Our findings are validated in numerical experiments.

翻译：本文研究旋转玻色-爱因斯坦凝聚体基态的数值逼近问题。该问题需要在黎曼流形 $\mathbb{S}$ 上极小化Gross-Pitaevskii能量 $E$。为求得相应的极小化子 $u$，我们采用基于Sobolev梯度概念的广义黎曼梯度法，并结合流形上自适应变化的度量。通过适当选择度量，可以证明该梯度法具有全局能量耗散特性。能量耗散性质进而保证了算法能全局收敛到 $E$ 在 $\mathbb{S}$ 上临界点 $u$ 的密度 $|u|^2$。此外，我们精确刻画了每个基态 $u$ 邻域内的局部收敛速率，并阐明该速率如何依赖于 $E^{\prime\prime}(u)$ 限制在 $u$ 的 $L^2$ 正交补空间上的第一谱间隙。由此，我们首次建立了旋转框架下极小化Gross-Pitaevskii能量泛函的黎曼梯度法收敛性结果。同时，我们改进了先前在无旋转情形下获得的研究结论。新分析中的主要复杂性在于极小化子的非孤立性——这些极小化子至多在复相位平移意义下具有唯一性。为此，我们在切空间 $T_{\mathrm{i} u} \mathbb{S}$ 中引入辅助迭代，并应用Ostrowski定理通过加权特征值问题刻画渐近收敛速率。随后，我们将辅助迭代与原始黎曼梯度法建立联系，并通过界定加权特征值问题的谱来获得定量收敛速率。数值实验验证了我们的研究结果。