Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Zixu Feng, Patrick Henning, Qinglin Tang

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 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.

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

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 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 -linear convergence rate and prove that the condition holds if and only if the energy sequence generated by P-RG converges locally -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.

Paper Structure

This paper contains 17 sections, 12 theorems, 138 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem settings and notations
  4. Properties of the problem
  5. Properties of the algorithms
  6. Sharp results on convergence and structure
  7. Main results
  8. The Morse--Bott condition for the finite classification of global minimizers
  9. Morse--Bott condition and optimal local linear convergence
  10. Beyond the Morse--Bott condition
  11. Technical lemmas
  12. Proof of main results
  13. Practical preconditioning under the Morse--Bott condition
  14. Numerical experiment
  15. Conclusion
  16. ...and 2 more sections

Key Result

Theorem 3.2

Suppose the energy functional $E$ satisfies the Morse--Bott condition in a neighborhood of each symmetry orbit Then the set of global minimizers $\mathcal{S}_g$ admits a well-defined classification.

Figures (2)

  • Figure 1: Contour plots of the density of the ground state $|\phi_g(\bm{x})|^2$.
  • Figure 2: (i) Error ratio $Q_{E^n} = \sqrt{(E(\phi^{n+1})-E(\phi_g)) / (E(\phi^{n})-E(\phi_g))}$ versus $n$ (red solid line). The black dashed line represents the theoretical rate $\rho_1$ predicted in Theorem \ref{['Opt-convergence']}. (ii) Error ratio $Q_{\phi^n} = \|\phi^{n+1} - \phi_g\|_{\mathcal{P}_{\phi_g}} / \|\phi^{n} - \phi_g\|_{\mathcal{P}_{\phi_g}}$ versus $n$ (red solid line). The black dashed line represents the theoretical rate $\rho_1$ predicted in Theorem \ref{['Opt-convergence']}.

Theorems & Definitions (29)

  • Definition 2.1: Morse--Bott Condition
  • Proof 1
  • Remark 2.5
  • Definition 3.1: Well-defined classification
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Lemma 3.7
  • ...and 19 more