# Two-dimensional Periodic Schr\"odinger Operators Integrable at Energy Eigenlevel

## TL;DR

This paper studies the geometric and topological properties of the Fermi curve of 2D periodic Schrödinger operators at zero energy, and constructs such operators using a generalized Novikov–Veselov method.

## Contribution

It demonstrates the smoothness and topological stability of the Fermi curve and develops a new construction method for these operators.

## Key findings

- Fermi curve is a smooth M-curve at zero energy
- Poles of Bloch solutions are on fixed ovals of an antiholomorphic involution
- Constructs operators with stable topological properties using a generalized Novikov–Veselov approach

## Abstract

The main goal of the first part of the paper is to show that the Fermi curve of a two-dimensional periodic Schr\"odinger operator with nonnegative potential whose points parameterize the Bloch solutions of the Shr\"odinger equation at the zero energy level is a smooth $M$-curve. Moreover, it is shown that the poles of the Bloch solutions are located on the fixed ovals of an antiholomorphic involution so that each but one oval contains precisely one pole. The topological type is stable until, at some value of the deformation parameter, the zero level becomes an eigenlevel for the Schr\"odinger operator on the space of (anti)periodic functions. The second part of the paper is devoted to the construction of such operators with the help of a generalization of the Novikov--Veselov construction.