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

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.

21 pages