Hidden $\mathfrak{u}(2,1)$ symmetry and Jordan chains in a resonant ghostly three-dimensional model

• URL: http://arxiv.org/abs/2606.02290v1
• Date: Mon, 01 Jun 2026 14:12:29 GMT
• Title: Hidden $\mathfrak{u}(2,1)$ symmetry and Jordan chains in a resonant ghostly three-dimensional model
• Abstract summary: We investigate a three-dimensional ghostly Hamiltonian realisation of the fully resonant sixth-order Pais-Uhlenbeck oscillator.<n>We construct intertwining operators whose quadratic combinations generate a hidden spectrum-generating $mathfraku (2,1)$-algebra.<n>We further derive a tri-Hamiltonian formulation from Lie point symmetries of the classical flow and show that the corresponding Hamiltonians are naturally encoded by the same hidden algebra.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We investigate a three-dimensional ghostly Hamiltonian realisation of the fully degenerate resonant sixth-order Pais-Uhlenbeck oscillator. On the classical level, the phase-space flow is non-diagonalisable and decomposes into two complex-conjugate Jordan chains of length three, explaining the appearance of oscillatory solutions with secular terms. Upon quantisation, we construct intertwining operators whose quadratic combinations generate a hidden spectrum-generating $\mathfrak{u}(2,1)$-algebra. The associated descendant spaces are finite-dimensional invariant subspaces carrying non-trivial Jordan structure. Although these spaces admit a natural decomposition into irreducible modules of a distinguished $\mathfrak{sl}_2$-subalgebra, this decomposition does not in general coincide with the Jordan decomposition of the Hamiltonian. We further derive a tri-Hamiltonian formulation from Lie point symmetries of the classical flow and show that the corresponding Hamiltonians are naturally encoded by the same hidden algebra. Nevertheless, unlike in the non-resonant case, no positive-definite linear combination of them generates the same dynamics. Finally, we analyse the common centraliser of the tri-Hamiltonian family in $U(\mathfrak u(2,1))$, showing that the natural higher-order candidate $Q$ is reducible and yields no independent classical or quantum integral. The model thus provides a resonant higher-derivative system in which hidden $\mathfrak{u}(2,1)$ symmetry, classical and quantum Jordan structures, and multi-Hamiltonian geometry coexist.

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