AI 中文总结
研究三维欧氏空间中两个相互作用的自旋-1/2粒子的量子超可积性，通过构建厄米一阶矢量算符，利用其与哈密顿量对易条件确定允许势及矢量积分，扩展了相关分类，还讨论了对称代数及标量约化的相关情况。
AI 中文摘要
我们研究三维欧几里得空间中两个相互作用的非相对论自旋-1/2粒子的量子超可积性。哈密顿量包含一个中心势以及自旋轨道、自旋-自旋、张量和二次自旋轨道相互作用项，所有这些都仅取决于相对距离。我们将分类限制在V4 = 0的情况下，即不包括自旋动量相互作用项。我们确定了所有允许非平凡一阶矢量运动积分的此类系统。为此，我们构建了由相对位置、动量、轨道角动量和两个自旋矢量组成的最一般的厄米一阶矢量算符。与哈密顿量的对易条件导致一个超定的径向确定方程组，其解给出了允许势的完整列表以及此类中的相应矢量积分。结果扩展了先前对两个自旋粒子的标量和伪标量一阶积分的分类。我们还讨论了由矢量积分生成的选定对称代数，并在一个代表性案例中展示了标量约化如何导致精确的库仑型和振子型解。
英文摘要
We study quantum superintegrability for two interacting non-relativistic spin-$\frac12$ particles in three-dimensional Euclidean space. The Hamiltonian contains a central potential together with spin-orbit, spin-spin, tensor, and quadratic spin-orbit interaction terms, all depending only on the relative distance. We restrict the classification to the $V_4=0$ case, so that the spin-momentum interaction term is not included. We determine all such systems admitting non-trivial first-order vector integrals of motion. For this purpose, we construct the most general Hermitian first-order vector operator built from the relative position, momentum, orbital angular momentum, and the two spin vectors. The commutativity condition with the Hamiltonian leads to an overdetermined system of radial determining equations, whose solution gives the complete list of admissible potentials and the corresponding vector integrals within this class. The results extend the previous classifications of scalar and pseudo-scalar first-order integrals for two particles with spin. We also discuss selected symmetry algebras generated by the vector integrals and show, in one representative case, how a scalar reduction leads to exact Coulomb- and oscillator-type solutions.
