The nonlocal Darboux transformation of the stationary axially symmetric Schrödinger equation and generalized Moutard transformation

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.