The nonlocal Darboux transformation of the stationary axially symmetric Schr\"odinger equation and generalized Moutard transformation

TL;DR

This paper introduces a nonlocal Darboux transformation for the stationary axially symmetric Schrödinger equation, generalizing the Moutard transformation, and provides new potentials and solutions as applications.

Contribution

It presents a novel nonlocal Darboux transformation that generalizes the Moutard transformation for the Schrödinger equation, with explicit formulas and new solutions.

Findings

Derived formulae for the generalized Moutard transformation.

Constructed new two-dimensional potentials.

Obtained exact solutions for the Schrödinger equation.

Abstract

The nonlocal Darboux transformation of the stationary axially symmetric Schr\"odinger equation is considered. It is shown that a special case of the nonlocal Darboux transformation provides the generalization of the Moutard transformation. Formulae for the generalized Moutard transformation are obtained. New examples of two - dimensional potencials and exact solutions for the stationary axially symmetric Schr\"odinger equation are obtained as an application of the generalized Moutard transformation.