The nonlocal Darboux transformation of the 2D stationary Schr\"odinger equation and its relation to the Moutard transformation

TL;DR

This paper explores a nonlocal Darboux transformation for the 2D stationary Schrödinger equation, establishing its connection to the Moutard transformation and generating new solvable operators with smooth potentials.

Contribution

It introduces a nonlocal Darboux transformation, relates it to the Moutard transformation, and provides new solvable Schrödinger operators with smooth potentials.

Findings

Special case of the nonlocal Darboux transformation yields the Moutard transformation.

New solvable 2D Schrödinger operators with smooth potentials are constructed.

The relation between two transformation methods is explicitly established.

Abstract

The nonlocal Darboux transformation of the two - dimensional stationary Schr\"odinger equation is considered and its relation to the Moutard transformation is established. It is shown that a special case of the nonlocal Darboux transformation provides the Moutard transformation. New examples of solvable two - dimensional stationary Schr\"odinger operators with smooth potentials are obtained as an application of the nonlocal Darboux transformation.