The Moutard transformation: an algebraic formalism via pseudodifferential operators and applications

TL;DR

This paper provides an algebraic formalism for the Moutard transformation, a 2D extension of Darboux transformation, with applications to spectral theory and the Novikov-Veselov equation.

Contribution

It introduces an algebraic interpretation of the Moutard transformation using pseudodifferential operators and extends the algebro-geometric formalism to 2D Schrödinger operators.

Findings

Algebraic interpretation of the Moutard transformation as conjugation.

Application to spectral theory of 2D Schrödinger operators.

Connection to the (2+1)-dimensional Novikov-Veselov equation.

Abstract

We consider the Moutard transformation which is a two-dimensional version of the well-known Darboux transformation. We give an algebraic interpretation of the Moutard transformation as a conjugation in an appropriate ring and the corresponding version of the algebro-geometric formalism for two-dimensional Schroedinger operators. An application to some problems of the spectral theory of two-dimensional Schroedinger operators and to the $(2+1)$-dimensional Novikov--Veselov equation is sketched.