Integrability of quantum dots

Abstract

We determine the frequency ratios $\tau \equiv {\omega }_z/{\omega }_{\rho }$ for which the Hamiltonian system with a potential \[ V=\frac{1}{r}+\frac{1}{2}\Big({ω_ρ}^2(x^2+y^2)+{ω_z}^2 z^2\Big) \] 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 ${\mathbb{R}}^3$.