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Frozen planet orbits for the $n$-electron atom

Stefano Baranzini, Gian Marco Canneori, Susanna Terracini

TL;DR

The paper tackles the existence of periodic frozen-planet orbits for an $n$-electron atom model constrained to a half-line with a fixed nucleus, formulating the problem variationally via the action $\mathcal{A}$ and addressing singular collisions with regularized functionals.A generalized Lusternik–Schnirelmann framework on manifolds with boundary, combined with a deformation-argument using a penalized functional $\mathcal{G}_{\lambda,c}$, yields at least one critical point, which is then shown to converge to a collisionless solution of the original system as the regularization parameter vanishes.The authors also analyze a zero-charge limit ($\mu\to0$) showing that $\mu$-frozen planet orbits converge to folded $nT$-brakes, linking the variational construction to Schubart-type brake orbits and providing insights relevant to highly ionized atoms.Overall, the work extends variational methods to a broad class of multi-electron, one-dimensional models with general attractive/repulsive potentials, offering a robust framework for constructing and understanding periodic atomic trajectories.

Abstract

We seek periodic trajectories of a system of multiple mutually repelling electrons on a half-line, with an attractive nucleus sitting at the origin. We adopt a variational viewpoint and study critical points of the associated Lagrange-action functional, by means of a modified Lusternik-Schnirelmann theory for manifolds with boundary. Additionally, when the charges of the electrons tend to zero, we show that frozen planet orbits converge to segments of a brake orbit for a Kepler-type problem, establishing a strong analogy with the Schubart orbits of the gravitational $n$-body problem.

Frozen planet orbits for the $n$-electron atom

TL;DR

The paper tackles the existence of periodic frozen-planet orbits for an $n$-electron atom model constrained to a half-line with a fixed nucleus, formulating the problem variationally via the action $\mathcal{A}$ and addressing singular collisions with regularized functionals.A generalized Lusternik–Schnirelmann framework on manifolds with boundary, combined with a deformation-argument using a penalized functional $\mathcal{G}_{\lambda,c}$, yields at least one critical point, which is then shown to converge to a collisionless solution of the original system as the regularization parameter vanishes.The authors also analyze a zero-charge limit ($\mu\to0$) showing that $\mu$-frozen planet orbits converge to folded $nT$-brakes, linking the variational construction to Schubart-type brake orbits and providing insights relevant to highly ionized atoms.Overall, the work extends variational methods to a broad class of multi-electron, one-dimensional models with general attractive/repulsive potentials, offering a robust framework for constructing and understanding periodic atomic trajectories.

Abstract

We seek periodic trajectories of a system of multiple mutually repelling electrons on a half-line, with an attractive nucleus sitting at the origin. We adopt a variational viewpoint and study critical points of the associated Lagrange-action functional, by means of a modified Lusternik-Schnirelmann theory for manifolds with boundary. Additionally, when the charges of the electrons tend to zero, we show that frozen planet orbits converge to segments of a brake orbit for a Kepler-type problem, establishing a strong analogy with the Schubart orbits of the gravitational -body problem.

Paper Structure

This paper contains 12 sections, 19 theorems, 195 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A critical point Theorem
  3. Assumption $ii)$ and $v)$ of Lemma \ref{['lemma:critical_point']}
  4. Palais-Smale conditions
  5. Assumption $vi)$ of Lemma \ref{['lemma:critical_point']}
  6. Proof of Theorem \ref{['thm:solution_helium']}
  7. Existence of $\varepsilon-$solutions
  8. Construction of $f_j^\varepsilon$, $\tilde{f}_j^\varepsilon$ and $\tilde{g}_{ij}$
  9. A priori estimates
  10. Convergence to a solution of \ref{['eq:helium_equation']}
  11. Zero-charge case
  12. Deformation Lemma

Key Result

Theorem 1

For any $T>0$ and any choice of $f_i,g_{ij}$ satisfying assumptions assumption:values_f_g-assumption:convexity, there exists a solution of eq:helium_equation such that:

Figures (2)

  • Figure 1: Two examples of frozen orbits provided by Theorem \ref{['thm:solution_helium']} for $n=4$ (above) and $n=3$ (below). In particular, this illustrates the qualitative behaviour for a solution of \ref{['eq:helium_mu']} when $\mu$ is small enough as in Theorem \ref{['thm:mu_to_0']}.
  • Figure 2: An illustration of the proof of assumption $vi)$ of Lemma \ref{['lemma:critical_point']}. Any capping of the $(n-2)-$dimensional sphere $\gamma_0$ defined in \ref{['def:gamma_zero']} must intersect the region $\{\mathcal{G}\le b\}$ in a point in which the action is bounded from below.

Theorems & Definitions (37)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Proposition 1
  • Lemma 2
  • proof
  • proof : Proof of Proposition \ref{['prop:no_solution_eigenvalue']}
  • Proposition 2
  • proof
  • ...and 27 more