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Abstract
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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 self-adjoint 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.
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Keywords
frozen planet orbits
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Mathematical Subject Classification
Primary: 47J30, 34C25
Secondary: 58B15
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Milestones
Received: 31 October 2022
Revised: 18 June 2023
Accepted: 4 July 2023
Published: 21 November 2023
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