在Tremblay { \sl et al}（2009 { \sl J. Phys. A：Math.Theor.}）的最近FTC中，{ \bf 42} 205206）中，证明了对于任意整数的$k$，平面上的某个新的精确可解可积量子哈密顿量$H_k$是超可积的，并且运动的附加积分是$2k$阶微分算子$Y_{2k}$ .本文证明了具有奇$k \ge 3$的Hamilton族$H_k$的猜想，其第一个成员对应于消除质心运动后的三体Calogero-Marchioro-Wolfes模型.我们的方法是基于一些$D_{2k}$ -扩展和不变的哈密顿量$\chh_k$，它可以被解释为一个修改的玻色子振子哈密顿量的建设。后者则具有$D_{2k}$不变的运动积分$\cyy_{2k}$，从其中$Y_{2k}$可以通过投影在$D_{2k}$身份表示空间。
英文摘要：
In a recent FTC by Tremblay {\sl et al} (2009 {\sl J. Phys. A: Math. Theor.} {\bf 42} 205206), it has been conjectured that for any integer value of $k$, some novel exactly solvable and integrable quantum Hamiltonian $H_k$ on a plane is superintegrable and that the additional integral of motion is a $2k$th-order differential operator $Y_{2k}$. Here we demonstrate the conjecture for the infinite family of Hamiltonians $H_k$ with odd $k \ge 3$, whose first member corresponds to the three-body Calogero-Marchioro-Wolfes model after elimination of the centre-of-mass motion. Our approach is based on the construction of some $D_{2k}$-extended and invariant Hamiltonian $\chh_k$, which can be interpreted as a modified boson oscillator Hamiltonian. The latter is then shown to possess a $D_{2k}$-invariant integral of motion $\cyy_{2k}$, from which $Y_{2k}$ can be obtained by projection in the $D_{2k}$ identity representation space.
The Tremblay-Turbiner-Winternitz system on spherical and hyperbolic spaces : Superintegrability, curvature-dependent formalism and complex factorization
Complex functions and geometric structures associated to the superintegrable Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion
Singular barriers and quartic integrability breaking in the TTW system
Universidad Autónoma Metropolitana Unidad Iztapalapa（墨西哥自治都市大学伊斯塔帕拉帕校区）；Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Reial Acadèmia de Ciències i Arts de Barcelona（巴塞罗那皇家科学与艺术学院）
