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Spectral and dynamical features for periodic/almost periodic coefficient sequences with 1/k^2 decay

Determine the spectral and dynamical features of the one-dimensional crystal with ν=1 and weight function w(k)=c_k/k^2 (extended symmetrically), when the coefficient sequence (c_k) is periodic or almost periodic with values in a finite set {0,1,...,p}.

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Background

Beyond random and generic settings, the authors ask for deterministic structure (periodic/almost periodic) in the selection of long-range edges with 1/k2 decay.

The goal is to understand how such structured choices influence spectral type and transport in these non-locally finite periodic graphs.

References

Problem 9.5. Consider the crystal over Z with ν = 1 defined by the weight function w(k) = c2 . Do periodic and almost periodic sequences of c kver {0,1,... ,p} have specific k spectral/dynamical features ?

— The curious spectra and dynamics of non-locally finite crystals (2411.14965 - Kerner et al., 22 Nov 2024) in Section 9, Problem 9.5