Generic spectral type of crystals

Characterize the generic spectral type (purely absolutely continuous, singular continuous, pure point, or mixtures thereof) for Schrödinger operators associated with crystals (Z^d-periodic graphs with finite fundamental cell and summable symmetric nonnegative edge weights), with respect to a natural topology on the defining data.

Background

The paper constructs examples realizing different spectral types in the periodic, non-locally finite class, including purely absolutely continuous and purely singular continuous spectra.

Motivated by this diversity, the authors pose a Baire-category type question about the spectral type that is typical (generic) among such periodic models.

References

Problem 9.2. What is the “generic spectral type” of crystals ?

— The curious spectra and dynamics of non-locally finite crystals  (2411.14965 - Kerner et al., 2024) in Section 9, Problem 9.2

We conjecture that a similar genericity result should hold for a broader class within tiling graphs, leading to generic Cantor spectrum, but the proof below uses the decorated $Z$-graph geometry in an essential way.

— Zero measure Cantor spectrum for aperiodic metric tiling graphs  (2610.02650 - Band et al., 2 Oct 2026) in Section 4, Subsection “Generic Cantor spectrum for decorated Z-graphs,” immediately before the proof of Theorem 4.4