Dice Question Streamline Icon: https://streamlinehq.com

Prove that the operators J_{2n} are conserved for all n

Prove that the operators J_{2n}=[J_{2n+1},a_0^†], with J_{2n+1} defined recursively by J_{2n+1}=[J_{2n−1},H] and J_1=a_0, commute with the Gérard–Grellier Hamiltonian H for all n, i.e., establish [H,J_{2n}]=0 as an analytic theorem.

Information Square Streamline Icon: https://streamlinehq.com

Background

Beyond the Lax-based construction, the authors empirically discovered an alternative tower of candidate conservation laws J_{2n} built via commutators starting from J_1=a_0. Numerical checks within individual (N,M) blocks suggest these operators commute with H.

An analytic proof that J_{2n} are conserved for all n would provide a rigorous foundation for this tower and strengthen the evidence for superintegrability of the quantum Gérard–Grellier model.

References

We have not been able to prove analytically that (\ref{J2n}) are conserved, though we have run extensive checks that the corresponding matrices commute with the Hamiltonian within individual $(N,M)$-blocks.

— A superintegrable quantum field theory (2511.03373 - Clerck et al., 5 Nov 2025) in Section “Quantum Lax pair and conservation laws” (around equations (oddJs) and (J2n))