Structure and symmetry of the Gross-Pitaevskii ground-state manifold
This provides a unified framework connecting geometric structure, symmetry, and algorithmic performance for researchers in mathematical analysis and computational physics.
The paper tackles the challenge of characterizing the ground-state manifold for the Gross-Pitaevskii energy functional under symmetries, establishing that the Morse-Bott condition partitions it into finitely many symmetry orbits and acts as the exact threshold for linear versus sublinear convergence in optimization algorithms.
The structure and degeneracy of ground states of the Gross-Pitaevskii energy functional play a central role in both analysis and computation, yet a characterization of the ground-state manifold in the presence of symmetries remains a fundamental challenge. In this paper, we establish sharp results describing the geometric structure of local minimizers and its implications for optimization algorithms. We show that when local minimizers are non-unique, the Morse-Bott condition provides a natural and sufficient criterion under which the ground-state set partitions into finitely many embedded submanifolds, each coinciding with an orbit generated by the intrinsic symmetries of the energy functional, namely phase shifts and spatial rotations. This yields a structural characterization of the ground-state manifold purely in terms of these natural symmetries. Building on this geometric insight, we analyze the local convergence behavior of the preconditioned Riemannian gradient method (P-RG). Under the Morse-Bott condition, we derive the optimal local $Q$-linear convergence rate and prove that the condition holds if and only if the energy sequence generated by P-RG converges locally $Q$-linearly. In particular, on the ground-state set, the Morse-Bott condition is satisfied if and only if the minimizers decompose into finitely many symmetry orbits and the P-RG exhibits local linear convergence in a neighborhood of this set. When the condition fails, we establish a local sublinear convergence rate. Taken together, these results provide a precise picture: for the Gross-Pitaevskii minimization problem, the Morse-Bott condition acts as the exact threshold separating linear from sublinear convergence, while simultaneously determining the symmetry-induced structure of the ground-state manifold. Our analysis thus connects geometric structure, symmetry, and algorithmic performance in a unified framework.