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arXiv · 2407.11191

Integrability of quantum dots

Maciej Dunajski·Andrzej J. Maciejewski·Maria Przybylska
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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 $\mathbb{R}^3$.

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Maciej Dunajski, Andrzej J. Maciejewski, Maria Przybylska. 2024-07-15. Integrability of quantum dots. https://arxiv.org/abs/2407.11191

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