Kolloquium der angewandten Mathematik

In this talk we investigate the numerical approximation of ground states of rotating Bose-Einstein condensates. This problem requires the minimization of the Gross-Pitaevskii energy functional on a Hilbert manifold. To find a corresponding minimizer, we use a generalized Riemannian gradient method that is based on the concept of Sobolev gradients in combination with an adaptively changing metric on the manifold. By a suitable choice of the metric, global energy dissipation for the arising gradient method can be proved. The energy dissipation property in turn implies global convergence to the density of a ground or excited state of the system. Furthermore, we show how the problem and the numerical method are linked to the solution of nonlinear eigenvalue problems through generalized inverse iterations. With this, we are able to present a precise characterization of the local convergence rates in a neighborhood of each ground state and how these rates depend on spectral gaps. Our findings are validated in numerical experiments.