Superintegrability of a resonant ghostly tri-Hamiltonian model: polynomial first integrals and transcendental completion
Abstract
We study classical and quantum superintegrability of a resonant three-dimensional Hamiltonian with indefinite kinetic energy and three Poisson descriptions of the same flow. Besides three commuting quadratic Hamiltonians, the polynomial first-integral ring contains two primitive degree-four generators, I4o and I4e, of odd and even momentum parity. A single octic relation leaves the polynomial invariant ring with transcendence degree four. Solving the complete first-integral equation shows that every rational first integral is a rational function of H1,H2,H3,I4o, so that the rational invariant field likewise has generic functional rank four. A branch-free transcendental invariant raises the generic smooth functional rank to five on the invariant open set Δ>0. The system is therefore minimally superintegrable in the polynomial and rational classes and maximally superintegrable on this invariant domain when transcendental integrals are admitted. Weyl quantisation yields exact higher-order differential symmetries in all three Poisson realisations. The branch-free fifth classical invariant also admits a densely defined nonlocal quantum counterpart, providing a concrete quantum realisation of the transcendental completion.
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BibTeX
@misc{fring2026superintegrability,
title = {{Superintegrability of a resonant ghostly tri-Hamiltonian model: polynomial first integrals and transcendental completion}},
author = {Andreas Fring and Ian Marquette},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01901},
eprint = {2610.01901},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01901},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Superintegrability of a resonant ghostly tri-Hamiltonian model: polynomial first integrals and transcendental completion
AU - Andreas Fring
AU - Ian Marquette
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01901
UR - https://arxiv.org/abs/2610.01901
AB - We study classical and quantum superintegrability of a resonant three-dimensional Hamiltonian with indefinite kinetic energy and three Poisson descriptions of the same flow. Besides three commuting quadratic Hamiltonians, the polynomial first-integral ring contains two primitive degree-four generators, I_4^o and I_4^e, of odd and even momentum parity. A single octic relation leaves the polynomial invariant ring with transcendence degree four. Solving the complete first-integral equation shows that every rational first integral is a rational function of H_1,H_2,H_3,I_4^o, so that the rational invariant field likewise has generic functional rank four. A branch-free transcendental invariant raises the generic smooth functional rank to five on the invariant open set Δ>0. The system is therefore minimally superintegrable in the polynomial and rational classes and maximally superintegrable on this invariant domain when transcendental integrals are admitted. Weyl quantisation yields exact higher-order differential symmetries in all three Poisson realisations. The branch-free fifth classical invariant also admits a densely defined nonlocal quantum counterpart, providing a concrete quantum realisation of the transcendental completion.
ER -