It is shown that planar quantum dynamics can be related to three-body quantum dynamics in the space of relative motion with a special class of potentials. As an important special case the O(d) symmetry reduction from d degrees of freedom to one degree is presented. A link between two-dimensional (super-integrable) systems and three-body (super-integrable) systems is revealed. As illustration we present number of examples. We demonstrate that the celebrated Calogero–Wolfes three-body potential has a unique property: two-dimensional quantum dynamics coincides with three-body quantum dynamics on the line at d = 1; it is governed by the Tremblay–Turbiner–Winternitz potential for parameter k = 3.

结果表明，平面量子动力学可以与具有特殊势能的相对运动空间中的三体量子动力学相关。作为一个重要的特殊情况，提出了从d自由度到一个度数的O（d）对称性减小。揭示了二维（超积分）系统和三体（超积分）系统之间的联系。作为说明，我们提供了许多示例。我们证明了著名的Calogero-Wolfes三体势能具有独特的性质：二维量子动力学与在线d = 1上的三体量子动力学一致；它由参数k = 3的Tremblay–Turbiner–Winternitz势控制。