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arXiv · 1702.01432

Rational integrability of trigonometric polynomial potentials on the flat torus

Thierry Combot
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Abstract

We consider a lattice $\mathcal{L}\subset \mathbb{R}^n$ and a trigonometric potential $V$ with frequencies $k\in\mathcal{L}$. We then prove a strong integrability condition on $V$, using the support of its Fourrier transform. We then use this condition to prove that a real trigonometric polynomial potential is integrable if and only if it separates up to rotation of the coordinates. Removing the real condition, we also make a classification of integrable potentials in dimension $2$ and $3$, and recover several integrable cases. These potentials after a complex variable change become real, and correspond to generalized Toda integrable potentials. Moreover, along the proof, some of them with high degree first integrals are explicitly integrated.

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Thierry Combot. 2017-02-05. Rational integrability of trigonometric polynomial potentials on the flat torus. https://doi.org/10.1134/s1560354717040049

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