Separability and Dynamical Symmetry of Quantum Dots

TL;DR

This paper explores the mathematical symmetries and separability of two-electron quantum dots, linking their internal dynamics to the perturbed Kepler problem and identifying classes of potentials that allow for analytical solutions.

Contribution

It identifies a broad class of axially symmetric potentials enabling separation in parabolic coordinates, extending previous work on quantum dot symmetries.

Findings

Established connection between quantum dot dynamics and perturbed Kepler problem.

Identified potentials allowing separability, including Stark and Hartmann potentials.

Provided detailed analysis of the harmonic trapping case.

Abstract

The separability and Runge-Lenz-type dynamical symmetry of the internal dynamics of certain two-electron Quantum Dots, found by Simonovi\'c et al. [1], is traced back to that of the perturbed Kepler problem. A large class of axially symmetric perturbing potentials which allow for separation in parabolic coordinates can easily be found. Apart of the 2:1 anisotropic harmonic trapping potential considered in [1], they include a constant electric field parallel to the magnetic field (Stark effect), the ring-shaped Hartmann potential, etc. The harmonic case is studied in detail.