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:我们研究三种预测的索博列夫梯度向地面流动的汇合情况,即Gross-Pitaevskii能源功能的梯度流动,相当于1美元-0美元(公吨)和2个其他等值的H美元(公吨)和1美元(公吨),包括自独立量(公吨)和自独立量(公吨)和自热量(公吨)。我们首先证明能源消耗特性和全球接近Gros-Pitaevskii能源临界点(离散时间为1美元(公吨)和零美元(升度),我们还证明所有三种计划都与地面状态本地指数趋同。