Applications of Cesàro summation to regularize winding numbers of unbounded-spectrum operators

Investigate whether Cesàro summation can be further employed to regularize winding numbers of operators with unbounded spectra beyond the Floquet Green’s function index N3[GF], and determine whether such regularizations yield physically meaningful, quantized topological indices or novel phenomena in additional physical systems.

Background

The paper establishes a generalized Středa framework for two-dimensional Floquet systems, connecting Floquet winding numbers to the magnetic response of the Floquet density of states. A key technical challenge arises from the anomalous spectral flow, which is represented by a formally non-convergent integral due to the unbounded Floquet spectrum; the authors regularize this quantity using Cesàro summation, obtaining a well-defined, quantized response.

This Cesàro regularization not only resolves the mathematical pathology but also provides physical insight: the anomalous flow corresponds to a quantized energy pump and to a quantized orbital magnetization density within a Floquet zone. The authors note they are unaware of other applications of Cesàro summation to regularize winding numbers for unbounded-spectrum operators, suggesting a broader mathematical and physical program to explore whether similar regularizations can characterize new topological indices and phenomena beyond the Floquet context.