Develop (S1) spectral covers for two-dimensional quasicrystal models

Construct algorithms that, for two-dimensional quasicrystal models (e.g., Laplacians on Penrose-type tilings and higher-dimensional Schrödinger operators), produce spectral coverings as finite unions of intervals that contain the spectrum and admit explicit, computable Hausdorff error bounds, thereby moving these classes from assumption (S2) to assumption (S1) in the Solvability Complexity Index hierarchy.

Background

Throughout the paper, the authors develop optimal algorithms for quantifying spectral size under two assumptions: (S1) access to spectral covers with controlled Hausdorff error and (S2) access to pointwise upper bounds on the distance to the spectrum. They show that many one-dimensional quasicrystal models admit (S1) covers, while higher-dimensional models typically only satisfy (S2), which requires an additional limit and hence higher SCI classification.

Lowering the SCI for two-dimensional models depends on developing (S1) spectral covers analogous to one-dimensional periodic approximations (e.g., via Floquet–Bloch theory). Achieving (S1) for two-dimensional quasicrystals would enable more powerful and efficient computations of spectral properties and facilitate computer-assisted proofs.