Structure and symmetry of the Gross-Pitaevskii ground-state manifold

TL;DR

This paper characterizes the geometric structure of the ground-state manifold of the Gross-Pitaevskii energy functional, linking symmetries, the Morse-Bott condition, and convergence rates of gradient methods.

Contribution

It provides a sharp theoretical description of the ground-state manifold structure, the role of symmetries, and the convergence behavior of preconditioned Riemannian gradient methods under the Morse-Bott condition.

Findings

Ground-state manifold decomposes into symmetry orbits under Morse-Bott condition.

Local linear convergence of P-RG occurs if and only if the Morse-Bott condition holds.

Failure of the Morse-Bott condition leads to sublinear convergence rates.

Abstract

The structure and degeneracy of ground states of the Gross-Pitaevskii energy functional play a central role in both analysis and computation, yet a precise characterization of the ground-state manifold in the presence of symmetries remains a fundamental challenge. In this paper, we establish sharp theoretical 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 intrinsic symmetries of the energy functional, namely phase shifts and spatial rotations. This yields a structural characterization of the ground-state manifold in terms of these symmetries. Building on this insight, we characterize the local convergence behavior of general preconditioned Riemannian gradient methods (P-RG). Under the Morse-Bott condition, we derive sharp local $Q$-linear convergence estimates 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 P-RG exhibits local linear convergence nearby. When the condition fails, we establish a local sublinear convergence rate. Taken together, these results show that the Morse-Bott condition is the exact threshold separating linear from sublinear convergence, while determining the symmetry-induced structure of the ground-state manifold, connecting geometry, symmetry, and convergence behavior in a unified framework.