On the nonlocal Darboux transformation for the stationary axially symmetric Schrödinger and Helmholtz equations

Abstract

The nonlocal Darboux transformation for the stationary axially symmetric Schrödinger 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 Schrödinger and Helmholtz equations are obtained as an application of the nonlocal Darboux transformation.