On a property of Fermi curves of 2-dimensional periodic Schr\"odinger operators

TL;DR

This paper investigates a specific geometric property of Fermi curves associated with 2D periodic Schrödinger operators, establishing a condition for the existence of solutions based on fixed points of an involution.

Contribution

It provides a necessary and sufficient condition for solutions related to Fermi curves of 2D periodic Schrödinger operators, linking geometric involution properties to divisor equations.

Findings

Solution exists iff the two marked points are the only fixed points of the involution.

Characterizes Fermi curves via divisor equations and involution fixed points.

Connects geometric properties of Riemann surfaces to spectral theory of Schrödinger operators.

Abstract

We consider a compact Riemann surface with a holomorphic involution, two marked fixed points of the involution and a divisor obeying an equation up to linear equivalence of divisors involving all this data. Examples of such data are Fermi curves of 2-dimensional periodic Schr\"odinger operators. We show that the equation has a solution if and only if the two marked points are the only fixed points of the involution.