A variational approach to frozen planet orbits in helium

TL;DR

This paper introduces a variational framework for identifying frozen planet orbits in helium atoms, utilizing regularization and mean-value interactions to establish their existence.

Contribution

It develops a novel variational approach with nonlocal functionals and mean-value interactions to prove the existence of frozen planet orbits in helium.

Findings

Existence of frozen planet orbits established

Use of Levi-Civita regularization with different time reparametrizations

Deformation of the helium problem to mean-value interaction model

Abstract

We present variational characterizations of frozen planet orbits for the helium atom in the Lagrangian and the Hamiltonian picture. They are based on a Levi-Civita regularization with different time reparametrizations for the two electrons and lead to nonlocal functionals. Within this variational setup, we deform the helium problem to one where the two electrons interact only by their mean values and use this to deduce the existence of frozen planet orbits.