Open Problems in Integrable Systems — Integrability and Quantisation

Definitions and mathematical statement

In the standard four-dimensional setting, a semitoric system is an integrable map $F=(J,H):(M^4,\omega)\to{\mathbb{R}}^2$ with $\{J,H\}=0$, where $J$ is a proper momentum map for an effective Hamiltonian circle action and all singularities are nondegenerate without hyperbolic components. A focus-focus point has the local Williamson quadratic pair, in suitable Darboux variables, \[ q_1=x\xi+y\eta,\qquad q_2=x\eta-y\xi. \] Let $c=F(m)$ be a focus-focus critical value. The source does not state whether the semitoric system is simple, in particular whether a singular fiber has only one focus-focus point.

For the usual single-focus setting, the Taylor series invariant is the formal Taylor expansion \[ \widehat S(X,Y)=\sum_{a,b\geq0}s_{ab}X^aY^b \] of a smooth regularized action germ, obtained by removing the universal logarithmic focus-focus singularity. Its precise representative requires the standard base-coordinate, cycle, orientation and additive/linear normalization choices. These conventions are not supplied in the record.

A quantum semitoric realization uses commuting semiclassical operators with principal symbols $J,H$. Its joint spectrum is a family $\operatorname{JointSpec}(\widehat J_\hbar,\widehat H_\hbar)\subset{\mathbb{R}}^2$ as $\hbar\to0$. S. Vũ Ngọc's task is to compute the Taylor series invariant directly from this spectrum "at" the focus-focus value. Thus the target is a reconstruction of the full formal invariant from spectral information localized at $c$, rather than merely identifying the singularity type.

The quotation marks around "at" are retained: the source does not specify which spectral neighborhoods or shrinking scales, semiclassical accuracy, multiplicities, labeling, or operator/subprincipal conventions are available. It cannot silently be read as reconstruction from a single eigenvalue or from a fixed-$\hbar$ point. Without those data and the invariant normalization, a unique reconstruction theorem or algorithm is not determined.

Short English statement

Recover every coefficient of a semitoric focus-focus Taylor invariant directly from joint spectral data localized at the critical value. The supplied assessment reports partial progress in quantum semitoric spectral theory but says a complete direct computation in that precise critical-value sense was not verified as finished. It gives no resolving or partial-result citation beyond the source problem, so this is an attributed verification limit rather than a definitive current-status claim.

Sources

• UnsolvedMath permanent ID 2100705, source code catalogue entry, "Open Problems in Integrable Systems --- Integrability and Quantisation." Registry curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0; updated 2026-08-06.
• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Problem 7.5, attributed to S. Vũ Ngọc: https://arxiv.org/abs/1804.03737. Supplied source-TeX extraction accessed 2026-07-29. The metadata says the source presents an open item and marks status/release and rights review NEEDS_REVIEW.
• The supplied partial-progress assessment refers generally to quantum semitoric literature without specific bibliographic entries. No missing references are invented.