Operator-theoretic foundations of the nonlocal quantum fifth symmetry

Establish a complete operator-theoretic treatment of the nonlocal quantum symmetry associated with the branch-independent fifth invariant, including the appropriate notion of independence required to formulate quantum maximal superintegrability.

Background

The paper constructs a branch-independent classical transcendental invariant Z_5 and a corresponding nonlocal quantum operator \widehat Z_5{(r)} for each of the three paired Hamiltonian realisations. The construction uses the inverse of a first-order operator and an exponential defined on a suitable dense class of analytic vectors, establishing a concrete conserved nonlocal operator rather than only a formal expression.

However, the construction does not provide a full analysis of the resulting unbounded operators. In particular, the maximal invariant domain, adjoint and spectral properties, and a rigorous notion of functional or operator independence from the polynomial symmetry algebra remain to be developed. Resolving these issues is necessary for a precise formulation of quantum maximal superintegrability.

References

A full operator-theoretic analysis, and in particular the appropriate notion of independence needed to formulate quantum maximal superintegrability, is left open.

— Superintegrability of a resonant ghostly tri-Hamiltonian model: polynomial first integrals and transcendental completion  (2610.01901 - Fring et al., 1 Oct 2026) in Section 1, Introduction; Section 6, Weyl quantisation and quantum symmetries