Further kinetic information from Floquet–Nevanlinna structure

Determine whether additional features of the Floquet–Nevanlinna analytic structure can encode further low-dimensional kinetic information and generate a sequence of progressively tighter spectral bounds between the geometric-mean bound and the exact spectrum.

Background

The paper constructs an analytic Floquet–Nevanlinna map whose behavior at large spectral parameters retains the geometric-mean kinetic scale, while its real-endpoint behavior recovers affinity–winding bounds. The authors explicitly ask whether other features of this same analytic representation can capture additional coarse-grained kinetic observables. A positive answer could produce a hierarchy of spectral constraints that progressively approaches the exact spectrum without requiring the full Markov generator.

References

Moreover, the analytic construction in our proof also suggests a broader hierarchy of spectral information. In the Floquet-Nevanlinna representation developed in this work, the large-spectral-parameter behavior retains the geometric-mean kinetic scale, whereas the real-endpoint behavior recovers the two reported affinity-winding bounds. This separation raises the question of whether additional features of the same analytic structure can encode further low-dimensional kinetic information and generate a sequence of progressively tighter bounds between the present result and the exact spectrum.

— Coarse-grained kinetic scale tightens thermodynamic spectral bounds of Markov cycles  (2608.22934 - Xu et al., 24 Aug 2026) in Section Discussion

Three matters are left open. The effective rate is defined but not evaluated, and its evaluation for a given adlayer is the point at which a model must be chosen. The hierarchy converges for the whole class, but how rapidly the two bounds close, and therefore how many moments are needed for a given accuracy, is a question about a particular spectrum, and the answer depends on the interaction and not on the class alone.

— The memory equation of a characteristic function in surface diffusion: a sum rule  (2609.31344 - Miret-Artés, 25 Sep 2026) in Section 7, Concluding remarks