Almost-sure spectral type and dynamics for random 1/k^2 weights

Determine, for the one-dimensional crystal with ν=1 and weight function w(k)=c_k(ω)/k^2 for k∈Z\{0} with i.i.d. random coefficients c_k(ω) on [0,1] (and c_{−k}(ω)=c_k(ω)), the almost-sure spectral type and transport properties of the associated operator H_Γ.

Background

The authors study non-locally finite crystals whose weights decay like k{-2} and observe markedly different spectral and dynamical behavior depending on which edges are present.

They propose randomizing the presence/strength of long-range edges (with 1/k2 decay) and ask for almost-sure results on spectral type and dynamics, while the overall structure remains periodic once a realization is fixed.

References

Problem 9.3. Consider the crystal over Z with ν = 1 defined by the weight function ck(ω) w(k) = k2 , where c kω), k ∈ N are i.i.d. random variables on 0,1 and c −k (ω) := ck(ω). Are there almost-sure spectral types and dynamics for the corresponding Γ?

— The curious spectra and dynamics of non-locally finite crystals  (2411.14965 - Kerner et al., 2024) in Section 9, Problem 9.3

These results leave open the distinction between pure point and singular continuous spectrum, but constitute a key step in proving Anderson localization in wave systems with critical long-range interactions.

— Absence of Absolutely Continuous Spectrum for One-Dimensional Capacitance Operators with Critical Long-Range Interactions  (2610.09582 - Ammari et al., 7 Oct 2026) in Abstract; Section 5, Concluding remarks