Extend the quantum G_n tower beyond G_3 and clarify its relation to the quantum Lax pair

Construct higher-order quantum conserved operators forming a tower analogous to the classical quantities G_n=Tr[ℒ^n], beyond the sextic operator shown in equation (G3q), and determine their relationship to the quantum Lax pair (\ref{defLMq}), including whether they can be systematically derived from that Lax construction.

Background

In addition to N and M, the authors provide a quartic conserved operator H_min and recover a sextic conserved operator (G3q) inspired by the classical Tr[ℒ3] structure. However, a general method to continue this tower is missing.

Clarifying how to extend this quantum G_n tower and how it ties into the proposed quantum Lax pair would unify the conservation-law framework and potentially yield a full hierarchy paralleling the classical integrable structure.

References

We do not know how to extend this tower to higher orders, nor is it clear what relation it bears to the quantum Lax pair (\ref{defLMq}).

— A superintegrable quantum field theory  (2511.03373 - Clerck et al., 5 Nov 2025) in Section “Quantum Lax pair and conservation laws” (around equation (G3q))

More generally, Eq.~eq:recursion shows that the full tower of moments ${O_{m,n}}_{m+n=2k}$ at fixed total degree $2k$ mixes among itself under time evolution via a finite-dimensional linear (in fact $\mathfrak{su}(1,1)$) representation, the harmonic-oscillator analogue of the free-streaming Calogero charges of Ref.; we leave a full classification of the resulting conserved combinations for general $k$ to future work.

— Exact Phase-Space Rotation in the Trapped Quantum Calogero Model  (2609.08987 - Sarkar, 8 Sep 2026) in Section 5.2, “Fate of the conserved charges”