Rigorous assessment of omitting the kick operator in NMR spectra

Characterize rigorously the impact of omitting the Floquet–Van Vleck kick operator on stroboscopic NMR spectra, and derive tight upper bounds for the resulting spectral deviations.

Background

The paper finds numerically that the kick operator can have a negligible effect on some stroboscopic NMR spectra, despite being required by the analytical Floquet framework for the exact transformed dynamics. The authors derive upper bounds for the spectral deviations caused by omitting the kick operator, but report that these bounds are not sufficiently tight. A comprehensive rigorous analysis of when and why the kick operator can be neglected therefore remains unresolved.

References

There are still open questions calling for further methodological research. One such issue concerns the convergence mechanisms of FME in general, but especially FVV, which is discussed but not yet fully solved with mathematical rigor. Based on our exemplary calculations and numerical validation routines, we empirically found that FVV provides the superior framework for an approximation both of the temporal evolution operator and of the resulting spectra. Yet, this is not mathematically proved by bounding the remaining deviations. Another question deals with the impact of the kick operator, which seems to be negligible in some NMR spectra for stroboscopic measurements. While we could provide upper bounds for the deviations in the spectra without kick operator these bounds are not very tight. A comprehensive, rigorous mathematical exploration is called for.

— Floquet Theory and Average Hamiltonian Theory Revisited: Equivalence, Convergence and Applications to NMR  (2609.17121 - Bock et al., 15 Sep 2026) in Section Conclusion and outlook