We study the convergences of three projected Sobolev gradient flows to the ground state of the Gross-Pitaevskii eigenvalue problem. They are constructed as the gradient flows of the Gross-Pitaevskii energy functional with respect to the $H^1_0$-metric and two other equivalent metrics on $H_0^1$, including the iterate-independent $a_0$-metric and the iterate-dependent $a_u$-metric. We first prove the energy dissipation property and the global convergence to a critical point of the Gross-Pitaevskii energy for the discrete-time $H^1$ and $a_0$-gradient flow. We also prove local exponential convergence of all three schemes to the ground state.

翻译：本文研究了三种投影Sobolev梯度流收敛到Gross-Pitaevskii特征值问题基态的性质。这些梯度流分别基于Gross-Pitaevskii能量泛函在$H^1_0$度量以及$H_0^1$上另外两种等价度量（包括与迭代无关的$a_0$度量及与迭代相关的$a_u$度量）的梯度流构造。我们首先证明了离散时间$H^1$梯度流与$a_0$度量梯度流的能量耗散性质，以及其全局收敛到Gross-Pitaevskii能量临界点的特性。此外，我们还证明了三种格式均具有局部指数收敛到基态的性质。