Frozen planet orbits for the $n$-electron atom

TL;DR

This paper investigates periodic electron trajectories in a one-dimensional atom model using variational methods, revealing convergence to classical Kepler orbits as electron charges diminish, and introduces a novel application of Lusternik-Schnirelmann theory.

Contribution

It develops a variational framework with modified Lusternik-Schnirelmann theory to find periodic orbits in a half-line electron system and connects these orbits to classical Kepler problem solutions.

Findings

Existence of periodic trajectories for electrons on a half-line.

Convergence of these trajectories to Kepler-type brake orbits as charges tend to zero.

Application of topological methods to a quantum-classical analogy.

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.