Quasiperiodic Structures and Non-Self-Adjoint Eigenvalue Problems in Material Sciences

We survey contemporary open problems in quasiperiodic materials and their spectral properties. Beginning with foundational toy models—such as the almost Mathieu operator, perhaps the simplest quasiperiodic operator exhibiting non-trivial spectral behavior, and the so-called Scottish flag operator—we explore key conjectures and unresolved questions where even numerical evidence is still lacking. We then review broader classes of quasiperiodic operators, highlighting how their spectral and dynamical properties influence the numerical approximability of their spectra. Finally, we turn to moiré materials, where quasiperiodicity plays a central role, and outline natural spectral questions that emerge in this rapidly developing field.