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.