Resolution sets and tractability of natural spectra

Determine whether natural spectra other than the harmonic oscillator admit nontrivial tensor resolutions and whether the resulting factorizations make the associated many-particle thermodynamics tractable, thereby characterizing how much of the harmonic oscillator’s solvability is explained by its resolution set.

Background

The paper shows that canonical thermodynamics depends only on the spectral alphabet, while different tensor factorizations can expose distinct combinatorial descriptions of the same spectrum. The harmonic oscillator is especially tractable because it admits infinitely many factorizations and geometric factors that allow principal specialization and permutation-statistics formulas.

The authors leave unresolved whether this combination of factorization freedom and computational tractability occurs in other natural spectra. This question concerns both the existence of analogous nontrivial resolution sets and the extent to which such resolutions account for solvability.