WKB structure in a scalar model of flat bands

Abstract: The scalar model of flat bands is a simplification of models in condensed matter physics. It allows the study of relevant spectral problems using a 2nd order scalar equation, akin to the Schroedinger equation with the square of dbar on a torus replacing the Laplacian. It displays many features of original models such as the ``quantisation" of the reciprocals of magic angles at which flat bands appear. The space of solutions can be described using a rank 2 holomorphic vector bundle over the torus and its properties as alpha varies are related to the structure of bands leading to a trichotomy: tangential touching (most of alphas), Dirac points (discrete set of alphas) and flat bands (discrete set). (Mengxuan Yang and Bryan Li observed that the same argument works in a more physically realistic setting of twisted two-layered wafers of graphene.) In my talk I will describe the basic properties of the scalar model and of the general class of scalar equations to which it belongs. I will also present a discussion of WKB-like structure of solutions. This is joint work with S Dyatlov and H Zeng, with earlier contributions by S Becker, M Embree, J Galkowski, M Hitrik, T Humbert, Z Tao, J Wittsten and M Yang.