On the convergence of Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem
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.
READ FULL TEXT