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

01/24/2023

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by   Ziang Chen, et al.

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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.