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

Ziang Chen; Jianfeng Lu; Yulong Lu; Xiangxiong Zhang

paper

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

arXiv:2301.09818

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 $H_{0}^{1}$ -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.