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arXiv · 2301.09818

On the convergence of Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem

Ziang Chen·Jianfeng Lu·Yulong Lu·Xiangxiong Zhang
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Abstract

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.

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Ziang Chen, Jianfeng Lu, Yulong Lu, Xiangxiong Zhang. 2023-01-24. On the convergence of Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem. https://arxiv.org/abs/2301.09818

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