平坦环面上三角多项式位势的有理可积性
Rational integrability of trigonometric polynomial potentials on the flat torus
摘要：
我们考虑格点$\mathcal{L}\subset \mathbb{R}^n$和频率为$k\in\mathcal{L}$的三角势$V$。然后，我们证明了一个强的可积性条件$V$，利用其傅里叶变换的支持。然后，我们使用这个条件来证明一个真正的三角多项式潜力是可积的当且仅当它分离到旋转的坐标。去掉实条件，我们还对$2$和$3$维的可积势进行了分类，并恢复了几种可积情形。这些势在经过复变数变化后变为实势，对应于广义Toda可积势。此外，在证明过程中，对其中一些具有高次首次积分的积分进行了显式积分。
英文摘要：
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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