Nondegeneracy and integral count of frozen planet orbits in helium

TL;DR

This paper proves the nondegeneracy and uniqueness of frozen planet orbits in helium and establishes their integral count as one, using advanced functional analysis and topological methods.

Contribution

It introduces a new proof of the nondegeneracy and uniqueness of frozen planet orbits, and computes their integral count as one through orientability and Euler characteristic techniques.

Findings

Critical points are always nondegenerate for the entire family.

Frozen planet orbit with mean interaction is unique.

Integral count of frozen planet orbits equals one.

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.