AI 中文总结
本文为受简谐势约束的量子Calogero模型构建了微观相空间描述，通过量子Lax对构造厄米Wigner算符，得到精确的相空间旋转演化方程，周期为2π/Ω，并验证了矩的旋转多重态性质及无陷阱极限的退化。
AI 中文摘要
我们发展了在外部简谐约束势存在下量子Calogero模型的微观相空间描述。基于量子Lax对结构，我们构造了一个厄米Wigner算符，其期望值对于任意初始态及相互作用强度的所有阶数都满足精确的相空间演化方程 d_t rho + lambda d_x rho - Omega^2 x d_lambda rho = 0。由此产生的动力学是相空间中周期为 2 pi/Omega 的刚性旋转，为受陷Calogero模型的等时动力学提供了微观实现。我们进一步表明，相空间密度的矩形成旋转多重态而非独立的守恒量。特别地，在二次扇区内，唯一的守恒组合正比于受陷哈密顿量，这为构造提供了非平凡的相容性检验。在 Omega -> 0 的极限下，该方程退化为无陷阱模型的精确自由流方程。
英文摘要
We develop a microscopic phase-space description of the quantum Calogero model in the presence of an external harmonic confining potential. Building on the quantum Lax-pair structure, we construct a Hermitian Wigner operator whose expectation value obeys the exact phase-space evolution equation d_t rho + lambda d_x rho - Omega^2 x d_lambda rho = 0 for arbitrary initial states and to all orders in the interaction strength. The resulting dynamics is a rigid rotation in phase space with period 2 pi/Omega, providing a microscopic realization of the isochronous dynamics of the trapped Calogero model. We further show that the moments of the phase-space density form rotating multiplets rather than independent conserved quantities. In particular, within the quadratic sector, the unique conserved combination is proportional to the trapped Hamiltonian, providing a nontrivial consistency check of the construction. In the limit Omega -> 0, the equation reduces to the exact free-streaming equation of the untrapped model.
CommentsI have withdrawn this submission to revise and further develop the work before considering a future submission
