On the nonlocal Darboux transformation for the stationary axially symmetric Schr\"odinger and Helmholtz equations

TL;DR

This paper develops a nonlocal Darboux transformation for stationary axially symmetric Schrödinger and Helmholtz equations, deriving formulas, relating it to the Moutard transformation, and generating new potentials and exact solutions.

Contribution

It introduces a novel nonlocal Darboux transformation, establishes its relation to the Moutard transformation, and provides new explicit solutions and potentials for these equations.

Findings

Derived formulas for the nonlocal Darboux transformation.

Established the relation to the generalized Moutard transformation.

Generated new two-dimensional potentials and exact solutions.

Abstract

The nonlocal Darboux transformation for the stationary axially symmetric Schr\"odinger and Helmholtz equations is considered. Formulae for the nonlocal Darboux transformation are obtained and its relation to the generalized Moutard transformation is established. New examples of two - dimensional potencials and exact solutions for the stationary axially symmetric Schr\"odinger and Helmholtz equations are obtained as an application of the nonlocal Darboux transformation.