The exactly solvable two-dimensional stationary Schr\"odinger operators obtaining by the nonlocal Darboux transformation

Abstract

The Fokker-Planck equation associated with the two - dimensional stationary Schr\"odinger 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 Schr\"odinger equation that provides the nonlocal Darboux transformation for the Schr\"odinger equation. This nonlocal transformation is applied for obtaining of the exactly solvable two - dimensional stationary Schr\"odinger equations. The examples of exactly solvable two - dimensional stationary Schr\"odinger operators with smooth potentials decaying at infinity are obtained.