Faddeev eigenfunctions for two-dimensional Schrodinger operators via the Moutard transformation

TL;DR

This paper explores how the Moutard transformation affects Faddeev eigenfunctions at zero energy for 2D Schrödinger operators, providing explicit examples for various potentials, including those related to the Novikov-Veselov equation.

Contribution

It introduces a method to analyze the impact of the Moutard transformation on eigenfunctions and offers explicit computed examples for different classes of potentials.

Findings

Moutard transformation modifies Faddeev eigenfunctions at zero energy.

Explicit examples for smooth, fast-decaying potentials are provided.

Connections to solutions of the Novikov-Veselov equation are demonstrated.

Abstract

We demonstrate how the Moutard transformation of two-dimensional Schrodinger operators acts on the Faddeev eigenfunctions on the zero energy level and present some explicitly computed examples of such eigenfunctions for smooth fast decaying potentials of operators with non-trivial kernel and for deformed potentials which correspond to blowing up solutions of the Novikov-Veselov equation.