Core property of the Koopman generator

Determine whether the closure of the explicit classical derivation on smooth cylinder functions coincides with the closed skew-adjoint generator of the Koopman group associated with the Gibbs-invariant renormalized Hartree NLS flow.

Background

The paper distinguishes between the explicit derivation defined by differentiating smooth cylinder functions along the renormalized Hartree NLS vector field and the closed skew-adjoint generator of the associated Koopman group on the Gibbs-space Hilbert space. The authors establish that the cylinder derivation is contained in the closed generator's domain and agrees with it on cylinder functions, but they do not establish that the cylinder functions form a core for the closed generator.

The remark identifies the unresolved issue with the analogous question for the dynamical Φ34\Phi^4_3 model: whether the closure of a classical gradient form equals the Dirichlet form of a process constructed by singular-SPDE methods. In the present setting, the corresponding question concerns equality between the closure of the classical Hartree derivation and the full Koopman generator.

References

We do not claim that _{\rm cyl} is a core for . This is analogous to the situation in , where the Dirichlet form associated with the dynamical \Phi4_3 model constructed by singular SPDE methods agrees with the classical gradient form on cylinder functions, while it remains open whether the closure of the classical gradient form coincides with the Dirichlet form of the constructed process.

— Dynamical classical-field limit of Bosonic Gibbs states: Renormalized Hartree NLS correlations in 2D and 3D  (2608.26477 - Nam et al., 27 Aug 2026) in Remark 2.4, Section 2.1

One may ask whether the integral paths obtained above can be identified directly with the Hartree NLS flow from Theorem~\ref{thm:canonical-flow}. In dimension three, constructs this flow as the limit of canonical finite-dimensional approximations by means of random-averaging operators. Theorem~1.3 and Remark~1.4 of show that several canonical approximation procedures give the same limit, but they do not state uniqueness among all paths satisfying the limiting Duhamel equation. The Schr\"odinger evolution has no smoothing effect that would automatically place a path furnished by the superposition principle in the random-averaging solution class. A direct appeal to would therefore require an additional weak--strong uniqueness statement showing that every dominated Duhamel path belongs to that class.

— Dynamical classical-field limit of Bosonic Gibbs states: Renormalized Hartree NLS correlations in 2D and 3D  (2608.26477 - Nam et al., 27 Aug 2026) in Remark 4.2, Section 4.1