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 H1 0 -metric and two other equivalent metrics on H1 0 , including the iterate-independent a0-metric and the iterate-dependent au-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 H1 and a0-gradient flow. We also prove local exponential convergence of all three schemes to the ground state.