Asymptotics for Fermi curves: small magnetic potential

TL;DR

This paper extends the analysis of complex Fermi curves associated with periodic Schrödinger operators to include small magnetic fields, characterizing their asymptotic behavior and topological properties.

Contribution

It introduces new results on the asymptotics of Fermi curves with small magnetic potential, generalizing previous work without magnetic fields.

Findings

Fermi curves are characterized in regions with large imaginary parts of coordinates.

Results show that generically these curves are Riemann surfaces of infinite genus.

The work extends the understanding of spectral properties of periodic Schrödinger operators with magnetic fields.

Abstract

We consider complex Fermi curves of electric and magnetic periodic fields. These are analytic curves in C^2 that arise from the study of the eigenvalue problem for periodic Schroedinger operators. We characterize a certain class of these curves in the region of C^2 where at least one of the coordinates has "large" imaginary part. The new results in this work extend previous results in the absence of magnetic field to the case of "small" magnetic field. Our theorems can be used to show that generically these Fermi curves belong to a class of Riemann surfaces of infinite genus.