We propose a new normalized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem based on an energy inner product that depends on time through the density of the flow itself. The gradient flow is well-defined and converges to an eigenfunction. For ground states we can quantify the convergence speed as exponentially fast where the rate depends on spectral gaps of a linearized operator. The forward Euler time discretization of the flow yields a numerical method which generalizes the inverse iteration for the nonlinear eigenvalue problem. For sufficiently small time steps, the method reduces the energy in every step and converges globally in $H^1$ to an eigenfunction. In particular, for any nonnegative starting value, the ground state is obtained. A series of numerical experiments demonstrates the computational efficiency of the method and its competitiveness with established discretizations arising from other gradient flows for this problem.

中文翻译：

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总的 Sobolev 梯度流--Pitaevskii 特征值问题：全局收敛和计算效率

我们为 Gross-Pitaevskii 特征值问题提出了一种新的归一化 Sobolev 梯度流，该问题基于能量内积，该内积依赖于流本身密度的时间。梯度流是明确定义的并收敛到一个特征函数。对于基态，我们可以将收敛速度量化为指数级快，其中速率取决于线性化算子的谱间隙。流动的前向欧拉时间离散化产生了一种数值方法，该方法推广了非线性特征值问题的逆迭代。对于足够小的时间步长，该方法会减少每一步的能量，并在 $H^1$ 中全局收敛到一个特征函数。特别地，对于任何非负的起始值，获得基态。