On complex Fermi curves of two-dimensional, periodic Schrödinger operators

The complex Bloch varieties and the associated Fermi curves of two-dimensional periodic Schrödinger operators with quasi-periodic boundary conditions are defined as complex analytic varieties, the Schrödinger potentials being from the Lorentz–Fourier space Fℓ∞,1. Then, an asymptotic analysis of the Fermi curves is performed. The decomposition of a Fermi curve into a compact part, an asymptotically free part, and thin handles, is recovered as expected. Furthermore, it is shown that the set of potentials whose associated Fermi curve has finite geometric genus is a dense subset of Fℓ∞,1. Moreover, the Fourier transforms of the potentials are locally isomorphic to perturbed Fourier transforms induced by the handles. Finally, an asymptotic family of parameters describing the sizes of the handles is introduced. These parameters are good candidates for describing parts of the space of all Fermi curves.