Integrability of quantum dots

TL;DR

This paper identifies specific frequency ratios for which a quantum dot system with a Coulomb and harmonic potential is completely integrable, linking it to conformal Killing tensors and geometric properties of the associated spacetime.

Contribution

It determines the integrability conditions for a quantum dot Hamiltonian and connects these to geometric structures like conformal Killing tensors.

Findings

Identifies frequency ratios for integrability.

Links integrability to conformal Killing tensors.

Shows particle trajectories are not geodesics of any Riemannian metric.

Abstract

We determine the frequency ratios $\tau\equiv \omega_z/\omega_{\rho}$ for which the Hamiltonian system with a potential \[ V=\frac{1}{r}+\frac{1}{2}\Big({\omega_{\rho}}^2(x^2+y^2)+{\omega_z}^2 z^2\Big) \] is completely integrable. We relate this result to the existence of conformal Killing tensors of the associated Eisenhart metric on $\mathbb{R}^{1, 4}$. Finally we show that trajectories of a particle moving under the influence of the potential $V$ are not unparametrised geodesics of any Riemannian metric on $\mathbb{R}^3$.