The Fokker-Planck equation associated with the two -dimensional stationary Schrödinger equation has the conservation low form that yields a pair of potential equations. The special form of Darboux transformation of the potential equations system is considered. As the potential variable is a nonlocal variable for the Schrödinger equation that provides the nonlocal Darboux transformation for the Schrödinger equation. This nonlocal transformation is applied for obtaining of the exactly solvable two -dimensional stationary Schrödinger equations. The examples of exactly solvable two -dimensional stationary Schrödinger operators with smooth potentials decaying at infinity are obtained.
The nonlocal Darboux transformation of the stationary axially symmetric Schrödinger 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 twodimensional potencials and exact solutions for the stationary axially symmetric Schrödinger equation are obtained as an application of the generalized Moutard transformation.
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