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Nondegeneracy and integral count of frozen planet orbits in helium

Kai Cieliebak, Urs Frauenfelder, Evgeny Volkov

TL;DR

The paper develops a comprehensive variational framework for frozen planet orbits in helium by introducing a continuum of functionals _r that interpolate between mean and instantaneous electron interactions. It proves universal nondegeneracy of all critical points along the interpolation, and, in the symmetric, normalized setting, establishes uniqueness of the symmetric frozen planet orbit for the mean interaction. A determinant-line orientation theory for self-adjoint Fredholm operators with spectrum bounded below yields a canonical Euler-characteristic count, which the authors show equals 1 for the relevant gradient sections, giving an integral count of frozen-planet orbits. The instantaneous-interaction case is handled similarly, with the same integral count result, and the work culminates in a robust variational-categorical framework connecting critical-point theory, elliptic-integral identities, and topological orientation data to produce a precise, integer-valued invariant of frozen-planet dynamics.

Abstract

We study a family of action functionals whose critical points interpolate between frozen planet orbits for the helium atom with mean interaction between the electrons and the free fall. The rather surprising first result of this paper asserts that for the whole family, critical points are always nondegenerate. This implies that the frozen planet orbit with mean interaction is nondegenerate and gives a new proof of its uniqueness. As an application, we show that the integral count of frozen planet orbits with instantaneous interaction equals one. For this, we prove orientability of the determinant line bundle over the space of selfadjoint Fredholm operators with spectrum bounded from below, and use it to define an integer valued Euler characteristic for Fredholm sections whose linearization belongs to this class.

Nondegeneracy and integral count of frozen planet orbits in helium

TL;DR

The paper develops a comprehensive variational framework for frozen planet orbits in helium by introducing a continuum of functionals _r that interpolate between mean and instantaneous electron interactions. It proves universal nondegeneracy of all critical points along the interpolation, and, in the symmetric, normalized setting, establishes uniqueness of the symmetric frozen planet orbit for the mean interaction. A determinant-line orientation theory for self-adjoint Fredholm operators with spectrum bounded below yields a canonical Euler-characteristic count, which the authors show equals 1 for the relevant gradient sections, giving an integral count of frozen-planet orbits. The instantaneous-interaction case is handled similarly, with the same integral count result, and the work culminates in a robust variational-categorical framework connecting critical-point theory, elliptic-integral identities, and topological orientation data to produce a precise, integer-valued invariant of frozen-planet dynamics.

Abstract

We study a family of action functionals whose critical points interpolate between frozen planet orbits for the helium atom with mean interaction between the electrons and the free fall. The rather surprising first result of this paper asserts that for the whole family, critical points are always nondegenerate. This implies that the frozen planet orbit with mean interaction is nondegenerate and gives a new proof of its uniqueness. As an application, we show that the integral count of frozen planet orbits with instantaneous interaction equals one. For this, we prove orientability of the determinant line bundle over the space of selfadjoint Fredholm operators with spectrum bounded from below, and use it to define an integer valued Euler characteristic for Fredholm sections whose linearization belongs to this class.
Paper Structure (23 sections, 26 theorems, 376 equations)

This paper contains 23 sections, 26 theorems, 376 equations.

Table of Contents

  1. Introduction
  2. Levi-Civita transformation
  3. The functionals $\mathcal{F}_r$ and their critical points
  4. Levi-Civita transform of critical points
  5. Analysis of critical points
  6. Proof of nondegeneracy
  7. Uniqueness of symmetric critical points
  8. Frozen planet orbits for mean interaction
  9. Relating frozen planet orbits to critical points of $\mathcal{F}_\rho$
  10. Proof of Theorem \ref{['thm:nondeg-av']} modulo two key lemmas
  11. Proof of Lemma \ref{['lem:const']}
  12. Proof of Lemma \ref{['lem:xi1xi2']}.
  13. Uniqueness of symmetric frozen planet orbits
  14. Determinant lines of self-adjoint Fredholm operators
  15. The determinant line bundle
  16. ...and 8 more sections

Key Result

Lemma 2.1

If $z\in C^0(S^1,\mathbb{R})$ has only finitely many zeroes, then the map $t_z:S^1\to S^1$ defined by eq:tz is a homeomorphism. $\square$

Theorems & Definitions (30)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Corollary 3.5
  • Lemma 3.6
  • Theorem 4.1
  • Lemma 4.2
  • ...and 20 more