Exact Phase-Space Rotation in the Trapped Quantum Calogero Model
Abstract
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.
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BibTeX
@misc{sarkar2026exact,
title = {{Exact Phase-Space Rotation in the Trapped Quantum Calogero Model}},
author = {Akash Sarkar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08987},
eprint = {2609.08987},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08987},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact Phase-Space Rotation in the Trapped Quantum Calogero Model
AU - Akash Sarkar
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.08987
UR - https://arxiv.org/abs/2609.08987
AB - 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.
ER -