Spectral determination of non-simple semitoric systems

Ascertain whether non-simple semitoric integrable systems are determined by the semiclassical joint spectrum of their associated quantum integrable systems; specifically, decide if recovery of the full set of non-simple semitoric invariants from the spectrum is possible or provide obstructions/counterexamples.

Background

For toric and simple semitoric systems, the inverse spectral problem has been resolved: their classical invariants can be reconstructed from the semiclassical spectrum, implying spectral determination via the known classifications.

The text notes that in the non-simple semitoric case (with multiple focus–focus points per fiber), such spectral determination is still unsettled; while expectations suggest a negative answer, a definitive result has not been established.