Product-closed subset of M for discrete systems with modulated couplings

Establish the existence and explicit characterization of a structured subset of M = {M ∈ SL(2, R) : |tr(M)| > 2} comprised of transfer matrices arising from one-dimensional periodic nearest-neighbour difference equations with modulated coupling coefficients (i.e., nonconstant coefficients on g(i+1) and g(i−1) in equations of the form a(i) g(i+1) + b(i) g(i−1) + V(i) g(i) = F(i, ω) g(i)), such that this subset is closed under finite products: for any finite sequence of matrices from the subset, their product also lies in M (equivalently, has absolute trace greater than 2).

Background

The paper studies band gaps in one-dimensional periodic systems by recasting the Floquet–Bloch spectral condition in terms of 2×2 unimodular transfer matrices. Band gaps correspond to matrices with |tr(M)| > 2, defining the set M ⊂ SL(2, R). The main theorem proves that for the special subset T = { [[0, -1],[1, t]] : |t| > 2 }, any finite product of matrices in T remains in M, enabling prediction of hierarchical band gaps in complex unit cells formed from simpler constituents.

Extending this closure property beyond on-site modulation (where nearest-neighbour couplings are constant) to systems with modulated couplings would broaden the applicability of the hierarchical band-gap principle. The authors explicitly conjecture that a suitably defined subset of matrices in M corresponding to such modulated-coupling systems can be identified so that products remain in M, mirroring the proven property for T.