Rigorous convergence theory for the Floquet–Magnus expansion

Establish mathematically rigorous convergence mechanisms for the Floquet–Magnus expansion in general, with particular emphasis on the Floquet–Van Vleck approach, and prove bounds on the remaining deviations of its approximations to the temporal evolution operator and resulting spectra.

Background

The paper develops the Floquet–Magnus expansion as a unifying framework for Average Hamiltonian Theory and the Floquet–Van Vleck approach, and reports numerical evidence that Floquet–Van Vleck approximations are generally more accurate and robust. However, the authors state that the mechanisms governing convergence, especially for Floquet–Van Vleck expansions, have not been fully resolved with mathematical rigor. In particular, the observed superiority of Floquet–Van Vleck approximations has not been established through rigorous error bounds.

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.

— 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